pases wd ee eRe ry ron Se ane oon S34slgerthneenermtarea tN woe, MON carey fore an grt ovens Sa Loe atte, ee ee ne peel sO ers Io TRANSACTIONS OF THE ROYAL SOCIETY OF VICTORIA, FROM JANUARY TO DECEMBER, 1860, INCLUSIVE. VOL V. Edited for the Council of the Society by JOHN MACGADAM, M.D., Hon. Sec. Zest ig. Midi lay / (s0eo8t. (te NOV1I3 936 a) Ns ah \ tne NAL mysewZ Melbourne: MASON AND FIRTH, PRINTERS, FLINDERS LANE WEST, ’ try at) ine { \ in) i yur Roval Society of Victoria. PRESIDENT : HIS EXCELLENCY SIR HENRY BARKLY, K.C.B, VICE-PRESIDENTS: RICHARD EADES, Esq., M.B. | PROFESSOR NEUMAYER. HONORARY TREASURER : REV. JOHN I. BLEASDALE. HONORARY SECRETARY : JOHN MACADAM, Esqa., M.D., M.L.A. COUNCIL: Cc. W. LIGAR, Esa. THE HON. DAVID WILKIE, M.D., JOUN MILLAR, Esq., C.K, : M.L.C. h ’ C. HODGKINSON, Esq., C.K. SOLOMON IFFLA, Hsa., M.R.C.S. 3 PROFESSOR IRVING. T. E. RAWLINSON, Esq., C.E. PROFESSOR M‘COY, WILLIAM GILLBEE, Esq., M.R.C.S.E. J. W. OSBORNE, Eso. FERDINAND MUELLER, Esq., M.D. H, F. EATON, Esa, CONTENTS. Inaugural Address of the President, His Excellency Sir Henry Arr. I. Il, Til. uA Ve VI. VII. Bark iy, K.C.B., &c., &e. The ‘*Three Sections,” the ‘‘ Tangencies,” tld aac ies Problem” of Apollonius, and Porismatic Developments, by Marrin GARDINER, C.E. Connery. Science peels Queen’s College, Galway) .. a A Communication from fie. Rey. W. B. Cr. LARKE, of Sydney, to His Excellency Sir Henry Barkty, K. C. B., &e., &c., President of the Royal Society of Victoria, on Professor M‘Coy’s “‘ New Teeniopteris ” from the Coal- bearing Rocks of the Cape Patterson District in par- ticular, , and on the Evidence bearing on the Question of the Age of Australian Coal Beds in general.—Commu- nicated to the Society by His Excellency THE PRESIDENT A Commentary on ‘‘A Communication made by the Rey. W. B. CrargkE to His Excellency Sir Henry Barxty, K.C.B., &e., &c.,-President of the Royal Society of Victoria, on Professor M‘Coy’s ‘ New Teeniopteris,’ &e., &e.,” by FrepErIcK M‘Coy, Esq., F.G.S., Hono- rary Fellow of the Cambridge Philosophical Society, Professor of the Natural Sciences in the University of Melbourne, Government Eolponteleeicis and Director of the National Museum +. The Conjugal Condition of the Pane of Victoria, con- sidered in relation to Laws of Divorce, by Francis A. Corser, Esq. : Remarks on the Physicai Guten A iy Climate, as of the Regions lying between the Rivers Lachlan and Darling, by W. Locknarr Morton, Esq. ... ee On the Nest and Eges of the Coach-whip Bird ( see Crepitans, Gould), : and of the White-fronted Epthianura (Epthianura Albifrons, Gould), with some general remarks on the Nidification of Australian Bir ds, by A. Dosrer, Esa. t =, the Application of G pe anic Electricity ¥o, Peete Astronomy, by R. J. L. Etiery, Esg., Government Astronomer a a PAGES 1—18 19—89 $9—95 96—107 . 108—127 128—140 141—144 . 145—152 vill Contents. PAGES Arr, VIII. Suggestions for the Introduction of Animals and Agri- cultural Seeds into Victoria, by W. LockHArt Morron, Esq. . = 5S . 153—157 LX. On a Deposit of Diatomaces at South es by Joun Coates, Esq., M.R.C.S.L., South Yarra... 158—164 X. Manners and Customs of the Agate Natives, in particular of the Port Lincoln District, by CHARLES Witueimy, Esq. “ac . 164—203 XI. Description of a New Species of Plumatella, ie 1B H. MacGinuivray, Esq., A.M., M.R.C.S. ... . 203—204 XII. Indigenous Fibrous Plants, i F. A. CorBert, mage 205—208 XIII. Remarks on Professor M‘Coy’s Commentary, by Rev. W. B. Cuarxke, M.A., F.G.S., &c. ae . 209—214 XIV. Note on Rev. Mr. Clarke’s in &e. ey Bee fessor M‘Coy ... . 215—217 XV. On the Multisection she an Fone by means of the Cycloid, by the Hon. D. E. WILKIE, MEADE, Mol witha plate ... ... 217—220 PROCEEDINGS. Minutes of the Meetings of the Society Sug we wee 1, —XXV. Annual Report for 1860 oe 306 de ve XXVil. —XXX. Balance Sheet for 1860 we nc as S53 XXX1,—XXxil, Reports of Committees e so: ae 0c XXXili.—xliy. List of Institutions, Learned Societies, &c., receiving copies of the Transactions of the Royal Society of Victoria for 1860 xly.—xlvii. List of Members ... are aie aa ae .. li, —bs. APPENDICES, containing— 1. Instructions to Members of Exploring Eepeition List of Stores, &c. Ixy. —lxxxiv. 2, Reprints of the Contents of former Volumes of the ‘* Transactions,” &e. pe we Sass Ixxxvil.—e. Minutes, &c., of Committee on Prize Essays ... oe end of volume Laws of the Institute oe wes wee tee ditto TRANSACTIONS OF THE Roval Society of Victoria. Inaugural Address of the President, His Excellency Sir Henry Banrxty, K.C.B., &c., &c. [Delivered to the Members of the Royal Society, at the Anniversary Meeting held on the 10th April, 1860.] GENTLEMEN or THE Roya Society or Vicroria, I am conscious that my unanimous election to the office on which I to-night enter, was in nowise owing to the scientific qualifications you supposed me to possess, but to your desire, as Members of the late Philosophical Institute, to evince a grateful sense of Her Majesty’s recent condescension, by thus making Her Representative your first President under the new style you have been permitted to assume. In this view I felt I had no right to decline the proffered compliment, nor to shrmk from the duties its acceptance B 2 Anniversary Address. imposed, however unequal I knew myself to be to their efficient discharge. ** Parcus scientiarum cultor et infrequens,” if I may be allowed so to paraphrase the confession of Horace, it is indeed with the greatest diffidence that I rise to deliver the Inaugural Address expected from me on this occasion, and to treat of scientific questions in the presence of many so much more conversant with their details than I can pretend to be. Most especially am I sensible of my unfitness to succeed one who has achieved so high a reputation in the scientific world as my predecessor in the Presidential Chair—Ferdinand Mueller—a man whose enthusiasm as a botanist is only sur- passed by his industry as a writer, evidenced, despite the official demands on his time, by the publication, during his presidency, of several most useful works, of which I need only instance his “ Monographs of the Tropical Eucalypti, of the Australian Acacias, and of the Genus Hremophila ;”” his «Enumeration of the Plants collected on the explormg expe- ditions of Gregory and of Babbage ;” his mvaluable “ Frag- menta Phytographie Australie,’ contaming already the diagnoses of no less than 600 new or undescribed Australian plants; and above all his first sheets of the “ Flora of Victoria,” a work which, when completed, will redound equally to the credit of the author, and of the Colony at whose expense it has been given to the world. As through his zeal and perseverance, moreover, the Philosophical Institute has, during the past year, obtained a “Jocal habitation,” as well as a change of name, and may now, therefore, be considered an established institution of the land, the present seems a fitting opportunity for me, when thus addressing you, to examine how far it has hitherto accomplished the objects for which it was founded, and to Anniversary Address. 3 consider by the hght of past experience how its operations may in future be most beneficially conducted. These objects were declared in 1855 (when two infant pro- jects of a similar nature were amalgamated into the present Society), “‘to embrace the whole field of science, with special reference to the development of the natural resources of the country,” the mode of effecting them having at the same time been defined to be ‘by original rescarches conducted by members, and original papers read and discussed at meetings.” I find that in accordance with the latter intention, not far short of one hundred Papers on points connected with Zoology, Botany, Geology, Astronomy, Meteorology, Enginecring, and other branches of science both exact and inexact, have been, during the five years of its existence, read and dis- cussed; and when I mention, not invidiously but for the sake of illustration, the names of Professors Wilson, McCoy, Hearn, and Neumayer; of Doctors Mueller and Becker ; of Messrs. Selwyn and Brough Smyth, as among the contribu- tors, I need hardly add that much has been thereby done for the advancement of scientific truth. I would not ignore the fact that the Society has been occasionally criticised by the lecal press for the trivial nature of some of the topics brought forward, no less than for the asperity which has at times characterised its discussions. Such criticism need occasion little concern. If well founded, it can scarcely fail to lead to improvement. If unfair, it will in the long run fall harmless. I am old enough to remember the attacks of the Times on the early proceedings of the British Association for the Ad- vancement of Science; but witty and vivacious as were its sallies, they aided probably rather than retarded the subse- quent triumphs of that noble scientific Congress. zy Anniversary Address. We are in fact all of us too prone to despise such pursuits as lead to no visible or immediate good to society; but, as has been well remarked, ‘‘no serious study is without serious value to the human race,” and a very little reflection will serve to show that the observation of facts cannot possibly be too minute or too accurate when we are engaged in inves- tigating the mysterious and all but inscrutable laws of Nature. Whilst I feel justified, however, in asserting that fair progress has been made towards realising one of the chief objects for which the Society has been instituted—namely, to elicit original communications on scientific subjects— I am, at the same time, free to confess that, looking to the many eminent names enrolled among its members, more, far more, of interest and importance might have been ex- pected to be contained in the three volumes of its Transac- tions which have as yet seen the light. My idea is, that the Society’s Transactions ought to present a faithful and enduring record of the position and progress of scientific research in this country, and should set forth the most advanced views that its savans entertain on the great scientific questions of the day, more especially those affecting the development of our material resources. From some of these gentlemen, however, mere passing and fugitive contributions have been obtained ; from others none at all; and so far from the Transactions reflecting the actual state of science in Victoria—of which there is, in my opinion, every reason to be proud—a stranger desirous of judging for himself on this point, would be obliged to gather scattered intimations of the labors of our scientific men from the reports of scientific bodies in Europe, from returns laid before the Legislature, or from even less accessible sources of information. i Anniversary Address. 5 To devise a remedy for this inadaquate representation of the state of science in our pages, may not be easy, but it will at any rate prove salutary to trace its causes. There is in Victoria, I need hardly remind you, no class of noblemen and gentlemen, as at home, devoting ample fortunes to the cultivation of such scientific pursuits as please them, and possessing abundant leisure to communicate their discoveries to one another. 4. It is a melancholy fact, though one almost incidental to the paucity of our population and the newness of our society, that neither literary nor scientific teaching will, except in connection with posts of public emolument, enable even the most talented to earn a livelihood in this country. I need not cite instances ; the experience of everybody will supply notafew. Hence it so happens that our leading scien- tific men have nearly all of them professional duties of more or less urgency to perform, and but little time at their dis- posal for the composition of papers not absolutely required by the State at their hands. So far from blaming them for this, we are bound to feel the deepest obligation to them for transferring their labors to our shores. There is scarcely one of them who would not have obtained higher honors if not larger emoluments in the Old World, and who does not therefore remain to explore the wide field opened at the Antipodes from pure love of science. It is natural that under such circumstances gentlemen should, when devoting their spare hours to describe the result of their observations, seek to influence as large a body of scientific readers as possible, and that they should, therefore, seldom address themselves to this Society, where the number of members who have turned their attention to any particular branch of knowledge is necessarily extremely limited. Hitherto, in fact, this paucity of members has constituted the great 6 Anniversary Address. obstacle to the introduction of a proper system of sectional division into our rules, and I fear no arrangement of the sort can work well until the number of cultivators of science is greatly augmented among us. At the last meeting of the British Association, at Aber- deen, upwards of two thousand members were present, fur- nishing material for full attendance, and an intelligent audi- tory im every section. Is it too much to anticipate that those throughout the Australian Colonies who take an interest in the advance- ment of science will, ere long, congregate in like manner in one or other of their Capital Cities every year ? We have intercolonial cricket matches for the development of the physical strength of our youth; champion races to test the fleetness of our horses. Why should we not have annual gatherings for the interchange of intellectual ideas ? Into such an arena it would be worth the while of our scientific men to descend. The Geologists might there satisfactorily discuss the theory of the distribution of gold in its matrix ; the Paleontologists determine whether our coal fields were really of the Paleeozoic or Mesczocic era, or whether secondary formations have any place on the Australian Continent or not. But until we can hold out an inducement in this way, we can hardly hope for such disquisitions on the different species of Ammonites and Belemnites as we had the pleasure of listening to from Professor McCoy on the occasion of Dr. Hochstetier’s recent lecture; nor be surprised that Mr. Selwyn should prefer to make known his theories through the journals of the Geological Society of Great Britain. There is indeed another way in which this Society, whilst more directly than heretofore fulfilling its avowed object of developing the natural resources of the country, might entitle itself to the gratitude, and perhaps enlist the services of the Anniversary Address. e heads of our scientific departments, and that is, by enforcing from time to time their applications for pecuniary aid upon the Government and the Parliament. I might, from the peculiarity of my public position, hesitate to refer more specifically to such a matter, had I not the high example of Prince Albert, who pointed this out to the British Association at Aberdeen, as one of the functions it was eminently fitted to discharge. T entirely concur with His Royal Highness in thinking that it is well for every country to possess a body capable of justly appreciating the wants of science, and of speaking with authority whenever the national interests demand the expen- diture of public money for scientific purposes. I am aware that this Society has, on some particular occasions, done so, but I think it might safely carry the principle further; and when it is clear, for example, that the printing of the maps of the Geological Survey is delayed for want of funds: that the collection of minerals and fossils made to illustrate it le useless, because neither space nor money is provided for their proper classification: that two or three hundred pounds would suffice for the publication of Meteorological and Astronomical Observations of the highest value made at the Flagstaff-hill Observatory: I hold it to be the duty of this Society to call public attention to the fact, and not to leave the officers concerned to supplicate for the requisite funds as if they had some private or personal in- terest to serve in obtaining them. I am confident that if such desiderata were fairly made known on competent authority, there would be no disinclina-— tion on the part of the Ministry to propose the vote necessary for their immediate attainment; and no one looking at the annual Estimates, and seeing the scale on which many of our scientific establishments are provided for, can accuse the Legislature of a niggardly disposition in such matters. Were 8 Anniversary Address. it otherwise, with the political constitution under which we live, it is not on either that the blame would fall. Lord Bacon, indeed, lays down the doctrine, that whatever the form of Government may be, “the state of knowledge is ever a democracy, and that prevaileth which is most agreeable to the senses and conceits of the people.” If there be then no just appreciation of the labors of scientific men; if there exist unreasonable impatience that the practical results of their investigations are not more rapidly realised: the fault is chargeable on the entire com- munity. I would not willingly imply that this is the case among us here. Every credit is due, I must repeat, to the Legislature for the liberality it has in times past displayed in scientific matters. It has cause, I have already recorded my belief, to be proud of the results achieved. For my own part, when I peruse the encomiums passed in presence of learned socie- ties at home, by so illustrious an authority as Sir Roderick Murchison, upon the labors of our Government Geologist and Palontologist ; or when I find so celebrated an astrono- mer as Mr. Hinds congratulating the scientific world, through the columns of the Times, “That the essential m- struments exist at Melbourne, in the hands of experienced Astronomers, for observing the transit of the newly discoy- ered intra-mercurial planet over the sun’s disc;” I feel, I confess, prouder of being the Governor of a colony which has attained this advanced stage of civilisation, than if I had a stately palace to dwell in, or barbaric hordes to bend the knee "ad Gy, GAL oe") If I am anxious, as I have hinted, to see still further moderate expenditure for scientific purposes; if I long to know that the Southern Heavens are nightly swept with an eighteen-inch lens, instead of by our present comparatively powerless telescopes ; it is because I am convinced that such Anniversary Address. 9 expenditure would in many respects be the truest economy, and that in others the fame which would accrue to the colony would far more than compensate for the immediate outlay. Our present position in regard to scientific researches strikes me as not very dissimilar to that of some quartz- crushing Company on our gold fields, possessing stacks of auriferous stone ready to yield untold treasure, together with a first-rate battery of stampers, but begrudging the fuel re- quisite for working the steam-engine by which that battery is to be driven. This Colony has gone to great expense in engaging the services of men of first-rate ability: it has provided each with certain apparatus suited to his vocation, or given him the aid of a costly staff. Having done this, will it be con- tent to rest in ignorance of what is beg accomplished, or delay, for the sake of a few extra thousands, the successful completion of their allotted tasks ? I have enlarged, gentlemen, at the risk of wearying you, upon this portion of my subject, because I am convinced that the Royal Society may do much to remedy such a state of things. It must, however, first of all, begin at the begin- ning, and acquire more of the confidence of the people than it at present commands. It must make science popular. Not in the false sense of that word. Not by patronising the exhibition of pretty tricks and ingenious experiments (though let me, in passing, observe I should be glad to hear lectures on scientific subjects, delivered by qualified lecturers, in our new Hall). Not by promoting the publication of cheap manuals and trashy guide books to science, in which facts are generalised until their individual significance is lost, and the student is persuaded that he can run before he really knows how to walk alone. Not by such means would I have you popularise the objects we have in view, but by teaching your fellow-colonists to believe in the earnestness of your purpose c 10 Anniversary Address. and the sincerity of your endeavours to promote their wel- fare ;. by inducing implicit reliance in the accuracy of your assertions, and inculcating the real value and utility of scientific truth. I know that difficulties await any society of this kind in such a course. Even among the educated classes there are many who deride the notion of a Philosophical Institute, and though not ignorant of the derivative sense of the words, nor unwilling to boast themselves “ friends of reason,” or posses- sors of “‘ knowledge,”’ would only apply the name of “ philo- sophers,” or ‘men of science,” to its members by way of taunt. That taunt is out of date, and futile. The revelations of the telescope on the one hand, and of the microscope on the other, so far from tending to exalt the pride of human reason, oppress man rather with a sense of utter msigni- ficance. We have no longer, as in Shakspere’s day, *¢Our philosophical persons to make modern And familiar things supernatural and causeless.” Such dogmatic half-knowledge, such pretension to superior wisdom, have long since vanished before the more general diffusion of education, and the humility inspired by a wider _ acquaintance with the boundless realms open to scientific research ; and we find in their stead that our scientific men are anxious and painstaking inquirers after truth, careful recorders of the facts their own special course of study may reveal. | Another class of doubters as to the utility of the Royal Society’s operations will be found in those who style them- selves “ par excellence” practical men. Very little consideration should serve to show these, how- ever, how enormously they benefit by the progress of scien- tific discovery, how ungratefully they too often appropriate its results with scarcely an acknowledgment. Anniversary Address. 11 Let any one who denies this read a pamphlet lately circu- lated, setting forth the claims of the Reverend Mr. Clarke, of New South Wales, in connection with the opening up of the mineral riches of Australia, wherein is clearly shown how valuable was this gentleman’s geological skill in directing the first miners aright ; how little even of the poor guerdon of thanks he has received from those who rushed afterwards to profit by his lessons. Or to cite a less familiar instance. Look at one of the greatest boons conferred on all in these Colonies—the shortening of the voyage between them and the Mother Country. Is this due, as might naturally be inferred, to the practical navigator? Was it effected by chance or rule of thumb? On the contrary, the credit belongs almost solely to Lieutenant Maury, of the United States Observatory at Washington, by means of whose Wind and Current Charts, in which the laborious records of innumerable voyages are compiled, the average passage to Australia was almost im- mediately reduced from 124 to 97 days. I might allude, if time allowed, to the Electric Telegraph, and more especially to the Submarine Cable, inventions which could never have extended their incalculable blessings to our very shores, had the Science of Electricity not been brought to its present advanced state by the labors and experiments of unremem- bered and unrewarded savans ; but I proceed to a third class of objectors to the study of Natural Philosophy, more diffi- cult to deal with still, because their objections are founded on a vague though conscientious apprehension that it leads to scepticism in matters of religious belief. This scruple is not new. It once extended even to the study of the Bible itself. Bacon found occasion to write— * Let no man, upon a weak conceit of sobriety, or an ill- applied moderation, think to maintain that a man can search too far, or be too well studied in the Book of God’s Words, or 12 Anniversary Address. in the Book of God’s Works; but rather let him endeavor at endless progress and proficience in both.” Yet, though silenced two centuries ago, it has of late, in consequence of the unexpected revelations of modern science, partially revived, and, if not often openly urged, creates strong pre- judice against the speculations of Geology, Astronomy, and other inductive pursuits, in the minds of many sincere Christians. To such I would with all respect submit—that any attempt to limit the scope of human inquiry must of necessity break down, whilst the mere desire to restrict it savors of want of faith in the truth of Divine Revelation. To reconcile God’s Works with God’s Words may, it is pos- sible, for ever transcend the faculties of mere humanity, but the believer may surely rest satisfied that what our present finite capacities cannot fathom, will one day be made clear, to all who have sought aright, in the mansions of eternity. Convinced, in the words of the laureate, “ that God and Nature are not then at strife,” let all echo the noble strain into which he bursts forth,— ‘¢ Who loves not knowledge? Who shall rail Against her beauty? May she mix With men and prosper! Who shall fix Her pillars? Let her work prevail !” But I must descend to less grand and exalted themes. To elevate the position of our Society to the extent I have indi- cated; to render it at the same time the depository of the opi- nions of the learned few, and the mouthpiece of the scientific wants of the unlearned many, are tasks for gradual accom- plishment. I have to allude to-night to one or two matters which I should like to see occupy your special attention during my presidential term. The first is the consideration of a scheme which has re- cently been proposed to the Government at home by my Anniversary Address. 13 enlightened compeer Sir William Denison, for the publi- cation of the Natural History, using the term im its widest significance, of the British possessions throughout the world, at the joint expense of the Mother Country and the Colonies, the latter contributing the descriptive materials and scientific data, the former being at the expense of engraving the illus- trations and editing the work. A more comprehensive or imperial design cannot easily be imagined, and I am happy to state it has received the cordial approval of the Royal Society of Great Britain, to whom it was referred for report by the late Secretary of State for the Colonial Department. Should it be determined to carry it into execution, the co-operation of this Society will no doubt be invoked, and from the able manner in which some of its members have recently reported, at the mstance of the Royal Society of Arts, on the productions of the Colony, I feel confident I may reckon on its aid to accelerate the scientific portion of the undertaking. Inthe meanwhile, it is worthy of being considered whether, in any event, certain principles of uniformity might not be introduced into all official publications in this and the adja- cent colonies, as has, I know, been attempted by the Regis- trar-General in regard to their statistics. A great deal of unnecessary trouble and expense, as well as of needless repeti- tion, might thus be spared in the production of any great National Work hereafter. There is a second subject of scientific interest which will I hope occupy a good deal of your attention during the ensuing twelvemonth—namely, the superintendence of preparations for the exploration of the interior. I rejoice that this Colony is at length about to take its share in this important National duty. Our very presence here to night may be said to be attributable to the love of science and of maritime discovery which the Anglo-Saxon 14: Anniversary Address. race has always displayed; for you will remember that it was to observe the transit of Venus over the Sun that Captam Cook was despatched on his first voyage to the Southern Ocean, and but for his subsequent exploration of the east coast of Australia, this continent might never have been colonised by the British nation. It seems but right, then, that this, the wealthiest and most civilised of the communities which have hence sprung into existence, should make some effort to advance the cause to which it owes its origin. Nor will that effort, I am per- suaded, prove fruitless. We can gain, it is true, no extension of territory by the discoveries we may make, yet no incon- siderable commercial advantages must ultimately accrue to the possessors of the only great haven on the south coast, from the progress of internal settlement, and from opening up a practicable route to the northern shores of this vast con- tinent; whilst a clearer insight into the nature and extent of the central desert cannot fail to elucidate phenomena now not easily explicable, and to complete our stores of informa- tion as to the Meteorology and Mineralogy, the Fauna and the Flora, of this most exceptional and extraordinary portion of the globe. To this Society belongs the honor of first directing atten- tion to the importance of such an expedition; it has subse- quently—stimulated by the munificence of the anonymous donor of £1,000—yraised a sum of nearly £3,000 towards this object, and has by its representations induced successive administrations to obtain from Parliament funds for the pur- chase of camels, as well as a further liberal erant of £6,000 to supplement the private subscription. The Society must, therefore, feel the deepest anxiety for the successful issue of an undertaking to which it thus stands | committed, and the Government has, in my opinion, acted wisely in resolving to leave its guidance and control to the Anniversary Address. 15 committee which has been appointed for the purpose, taking care, of course, that nothing is done without its knowledge, and that proper checks are imposed on the issue and disburse- ment of the money voted. The committee has prudently decided that nothing shall be attempted during the approaching winter, which would have been too far spent ere the exploring party could have reached its starting poimt; but I trust that every pains will be taken in the spring to organise and equip an expedition worthy of this colony, and that by the commencement of the ensuing summer it will be on its way, under a leader of approved ability, to the depét selected upon Cooper’s Creek as the basis of its operations, so as to be ready to take advantage of the first rains that may fall, to prosecute its researches. The precise direction of these must necessarily be left a good deal to the discretion of the leader to be chosen. Were not something more than a mere bush ride across the conti- nent aimed at, it might be the easiest course to proceed at once to the westward of Lake Torrens, where that daring veteran Stuart, and my no less gallant friend the Governor of South Australia, have already penetrated country which seems to promise a passage to the north. My own opinion has, however, always been in favor of directing the earlier efforts of the expedition to ascertaining the exact eastern limits of the Great Desert, with a view to crossing as directly as possible to the Gulf of Carpentaria, or to Arnheim’s Land, the great promontory by which the western shore of that gulf is formed. These, gentlemen, are the special questions on which I am chiefly desirous that your immediate attention should be bestowed. You will not, I feel sure, suffer them to interfere with the zealous discharge of your ordinary duties as members of the Royal Society, but will, on the contrary, devote yourselves 16 Anniversary Address. with redoubled ardor to the task of rendering our monthly meetings profitable and agreeable. A noble field lies before us. There is ample room for all! Let every one set earnestly to work in his own sphere for the advancement of science ; he who never did so before taking up some branch in which more accurate knowledge is still desirable. Let those who find aught worthy of being communicated favor us with papers, to be discussed with moderation of language, and in entire oblivion of bygone bick- erings. Let this be done, and we shall hardly fail to achieve results of importance; for, in the words of an eloquent writer on natural history in the Cornhill Magazine, “from the illu- mination of many minds on many points, truth must finally emerge.” Association for scientific research is, in fact, no longer matter of choice, but of necessity. The collection and classification of facts is the essential element of modern pro- gress, and it cannot be attained without division of labor and widespread publicity. In earlier stages of the world’s his- tory the brain of a single man, of an Aristotle or a Pliny, sufficed to comprehend all that was yet discovered regarding Nature, but such knowledge is now too vast to be grasped in sufficient clearness of detail by any individual intellect. Hyen the giant mind of Humboldt quailed before the task of giving a physical description of the universe, and confessed the com- pletion of his “Cosmos,” according to his original concep- tions, to be impracticable. True genius is indeed ever humble. The great Newton described himself towards the end of his career as having only gathered a few pebbles on the shores of a boundless ocean. Who in our day shall venture to boast of domg more than sift some grains of the sand which brims that ocean’s shores? May we united pursue the path of scientific inquiry, in a like spirit of humility, and with an eye to truth alone. Anniversary Address. Wy ** Let knowledge grow from more to more, Yet more of reverence in us dwell.” May we, like them, whilst fearlessly scrutinising Nature’s laws, cease not for a moment to respect the teachings of in- spiration, nor forget to look from Nature up to Nature’s God. Long as I have already trespassed on your time, I cannot adequately give utterance to the feelings which I entertain on this head, without, in conclusion, adopting, in its integrity, the impressive language of one of the greatest orators and divines the New World has yet produced, Dr. Channing :— “I look with admiration on the intellectual force which combines and masters scattered facts, and by analysis and comparison ascends to the general laws of the material universe. But the philosopher who does not sce in the force within him something nobler than the outward nature which he analyses—who, in tracing mechanical and chemical agencies, is unconscious of a higher action in his own soul— who is not led by all finite powers to the Omnipotent, and who does not catch, in the order and beauty of the universe, some glimpses of spiritual perfection—stops at the very threshold of the temple of truth.” 19 Art. I.—The “ Three Sections,’ the “ Tangencies,’ and a “Loci Problem” of Apollonius, and Porismatic Develop- ments. By Martin Garpiner, C. E, (formerly Science Scholar, Queen’s College, Galway). {Read before the Royal Society, June 4, 1860, ] PRELIMINARY OBSERVATIONS. In the Transactions for 1859 I promised solutions to the celebrated problems of the Greek and French schools, and the present paper is the first instalment towards the fulfil- ment of that promise. I commence with the problems of Apollonius, known as his “Three Sections,” and “ Tangencies,” and the principal problem of his treatise on Loci; but I propose also the con- tinuation of the development of interesting ‘‘ Porisms.” The problems of the Three Sections are famous from the number of geometers who have assayed their solutions. Willebrord Snel (the first person who measured the length of an arc of the meridian by means of a geodetic survey), who was born at Oudewater, in Holland, in the year 1590, was the first geometer of eminence to restore the Section of Ratio. His solution was published at Leyden, in 1608. Early in the eighteenth century, Dr. Halley discovered an Arabic manu- script in the Bodleian Library containing distinctinvestigations to the numerous subdivisions of the Section of Ratio, a Latin edition of which he published at Oxford in the year 1706; but there is no evidence as to whether this relic is a transcript from the original of Apollonius, or merely a string of solutions to its various cases by some other geometer; it covers 138 pages. Since then the principal solution is that by Reuben Barrow, which was published about the year 1780 in his “ Apollonius.” An application of the problem may be seen in David Gregory’s Astronomy. The Section of Space received an original solution from Dr. Halley, which is similar to that given in Leslie’s Geome- trical Analysis. Other solutions may be found scattered through mathematical periodicals ; but as they are all similar and incomplete, they deserve no particular notice. Indeed, an unaccountable neglect has been shown to this problem by the geometers who attempted the other “ Sec- c 20 The Three Sections, Tangencies, tions;”’ and this is the more strange as the Section of Space is by far the most useful of the three. The Determinate Section was solved by Willebrord Snel, and since then by Dr. Robert Simson, Wiliam Wales, and Petro Giannini. Snel’s and Wales’ solutions were re- published at London in 1772, by the Rev. John Lawson, and Giannini’s at Parma in 1773. Dr. Simson’s solution was published in his Opera Reliquain 1776, at the private expense of Earl Stanhope, and covers over 150 pages. However, though the lost writings of Apollonius occupied the attention of Newton, Halley, Simson, Burrow, Huygens, D’Omerique, Lalouere, and a host of other distinguished geometers, it is a most remarkable fact that none of them perceived the liaison of “The Three Sections.” Indeed, it was only through the instrumentality of the Homographic Theory, as systematised by M. Chasles, Pro- fessor of Geometry to the Faculty of Sciences of Paris, that this intimate connection was exposed, and analogous solutions for the first time given. Chasles’ solutions—extracted from his correspondence with the late Professor Davies, of the Royal Military Academy—dated 1848, were published in the third volume of the Mathematician, and again in his recent work entitled Traité de Géométrie Supérieure. These latter solutions are more in detail than those in the Mathematician, and the following accompanying observations of the author, who has been justly styled the Newton of Geometry, are worthy of special attention. He says :— ‘“‘Amongst thenumerous questions to which the homographic theory can be most easily applied, are those which formed the subject of the three works of Apollonius, entitled the Section of Ratio, the Section of Space, and the Determinate Section. Hach of these questions exacted a great number of propositions. Pappus relates that there were 181 in the Section of Ratio, 124 in the Section of Space, and 83 in the Determinate Section. These arose from the fact that the solution to the general question was never given directly, as the ancient geometers proceeded to first establish the most simple cases, and then went step by step to the more general, so that the solution of each case always depended on those which preceded. Moreover, each problem gave rise to as many different questions as there were varieties in the different relative parts of the figure. _ In the two last centuries these problems have occupied the attention of many eminent geometers, who endeavoured to and Loci of Apollonius, &c. 21 restore the works of Apollonius; but, although they tried to reduce the solution of each to as few propositions as possible, it is yet the same long and tortuous method they have all followed. For instance, J. Leslie gives four propositions to the Section of Ratio, six to the Section of Space, and eight to the Determinate Section, whilst, by my method, one solution suffices for the three questions, considered in their most general forms.” Now, I have already recorded my opinion concerning the peculiar method of investigation of the ancient geometers and their modern imitators, namely, that it is attributable to the want of precision and generality in the indicated operations, and involved theorems ; but I will further observe, in this place, that the homographic theory must receive some developments in limits to the constants of the equations, implicating the double points of divisions on the same straight line, before it becomes thoroughly effective in its applica- tions. And from the absence of such developments, Chasles’ solutions are necessarily defective. Take for instance his solution to the Section of Ratio, which is as follows* :— “Draw AE parallel to NN, to cut MM in E; draw AG parallel to MM to cut NN in G; find I in MM such that PI: RG :: m: n; bisect IE in O; in NN find H such that PO : RH :: m:n; draw HA to cut MM in F; from O as centre and radius = (OF:OE)? describe a circle ; through either point C in which this circle cuts MM, draw CA to cut NN in D: then will CAD be an answerable line.” And his only remarks in respect to the limits of the problem are—‘‘ And if the segments OF and OE be not on the same side of O, the two solutions will be imaginary.” Here it is evident that when the given straight lines MM, NN, are parallel, the method is not intelligibly applicable. And it is but right to observe that this is the only case in which the principal construction given in Leslie’s Geometrical Analysis (introducing the improvement of indicating opposite directions by opposite signs) cannot be applied. However, the general method of finding the double points of homographic divisions which is given in the Géométrie Supérieure, would, if introduced, overcome this imperfection. But there is a much more serious defect which cannot be rectified by the “theory,” such as it now exists, namely, the non-establish- *See the enunciation I give to this problem. 22 The Three Sections, Tangencies, ment of the precise limiting values for the ratio=. Surely, it is not evident that there are two limiting positions for F, such asf and f’, and that according as the ratio = lies outside the limits £4 and 74; (h and fh’ being the points in which fA and f’A cut NN), or is equal to one of them, or is com- prehended between them, so will the corresponding points C be real and distinct, real and coimcident, or imaginary. __ And similar remarks apply to his solutions to the Section of Space and to the Determinate Section ; for the homogra- phic theory will not establish the /imits, nor even hint as to their nature or number. My solutions are equally general with those given by Chasles, and—as will be seen in the Generating Problem— one wording applies to the three questions in their most general forms. Besides, they possess the distinguishing characteristic of being intelligibly applicable to all the parti- cular cases; and the simple considerations, by means of which the limits are established, will be found applicable to the determinations of limits in numerous other important questions. The next in order of the works of Apollonius, after the Determinate Section, was the “ Tangencies.”’ The enunciation of the problem, and some of the “ Lemmas’”’ used in its solution, which were preserved in the Mathemati- cal Collection of Pappus, enabled Dr. Robert Simson, of Glasgow, to reproduce one case (that of two circles and a point) though not under its original form,—as may be seen in the Appendix to his Opera Reliqua; but a more elegant solu- tion to the same was previously given by Vieta, in his Apollo- nius Gallus. And since Dr. Simson’s, an entirely different solution has been given by Monsieur Auguste Cauchy, in the “Correspondence de |’Ecole Polytechnique.” However, neither Simson, Vieta, nor Cauchy sueceeded in giving a direct solution to the general question. Newton virtually solved the general question in his Prin- cipiz, where it entered into some astronomical determina- tions ; and, indeed, it is the only direct geometrical solu- tion by a British geometer which applies to the various cases, when we suppose the circles to have any value from zero to infinity. But the most complete and elegant soluticn hitherto given to the “‘Tangencies,”’ is that of M. Gergonne, in the Annales de Mathematique, which (according to M. Chasles) is an and Loci of Apollonius, &c. 23 improvement on a solution by M. Gaultier, in the Journal de VEcole Polytechnique. In this paper I give ten direct geometrical solutions to the general question. The first of these is, I consider, the simplest ever given Its applications to the case in which two of the circles are finite, and the other circle infinitely small, is an improvement on Vieta’s solution ; and to the case where two of the given circles are infinitely small, and the third finite, it is similar to what is given by Brianchon* in the Journal de l’Ecole Poly- ~ technique. The second solution is also applicable to all the cases of the problem; and the idea of the auxiliary circle can be applied in other questions, so as to render the solutions more general. The application of the third solution to the case, in which two of the given circles are finite, and the third infinitely small, leads to M. Cauchy’s method for this case, &c. The other solutions are applicable to all the leading cases of the problem, but fail to indicate graphical constructions for some of the minor ones, owing to the peculiarities inherent in the involved theorems, or in the methods of contemplating or expounding them. The tenth is most probably a reproduc- tion of Apollonius’ solution. The “Loci Problem,” which I have undertaken, is ina more general form than was accorded to it by Apollonius. It comprehends almost the entire substance of the Benoa Book, as restored by Dr. Simson. The solution is direct and general; besides, it shows that when the ratio is unrestricted in sign, the locus is not (as usually intimated) @ circle, but two real circles, a real circle and a point, or a real circle and an imaginary one, according to relative states of the data. Particular cases only of this problem were solved by Dr. Simson, all of which have been republished in Leslie’s Geo- metrical Analysis. His methods are inapplicable to the general question, as they depend on the reality of a point in the straight line passing through the given points, which may be- come imaginary, even when the locus is real. A method of constructing the locus, having many points approaching to mine, is given in the Geometry of the Library of Useful Knowledge; but there, too, the process depending on points which may be imaginary when the locus * Professor Davies has erroneously confounded Brianchon’s with Pappus’ solution. —(See vol. 3, page 227, Mathematician. ) 24 The Three Sections, Tangencies, is real, is applicable only in particular states of the data. In the Notes will be found a genuine ancient porism, from which the problem originated. I might also mention that Francis Van Schooten, Professor of Mathematics at Leyden, published a restoration of some of the particular cases of this problem, in 1657 ; and that a like task was performed in an algebraic form by Fermat, Coun- cillor to the Parliament of Toulouse, in his Opera Varia Mathematica, published in the year 1679. It is scarcely necessary to remark, that in the present improved state of Algebraic Geometry, it would bean easy matter to solve the general case of this problem; but, to arrive at a construction of the Locus, such as I give, would be impossible without introducing other geometrical considera- tions than those to be found in ordinary Algebraic Elements ; besides, the complete discussion would present difficulties which none but experienced analysts could overcome. | The ‘‘Porisms” in the present paper, with those in the Transactions for 1859, belong to the most numerous and useful system in the whole range of elementary theorems. Some few of them—as is evident from Professor Davies’ con- tributions to the Mathematician—have been already noticed by Mr. Mark Noble, and by Professors Playfair and Wallace ; but their number is so few, that when they occur I will make no scruple of reproducing them amongst the classes to which they belong. I have already given proof of their efficiency in the solution of difficult problems. They are most probably but restitutions of a part of the lost treatise of Euclid, known as his Second Elements—com- posed when his geometrical knowledge was fully matured, and which, there is strong reason to suspect, contained all the principles developed in the elementary writings of Gergonne, Poncelet and Chasles. Having said thus much relating to the substance of my paper, I think it’right, before closing these prelimimary re- marks, to explain the nature of the improved ideas and theorems on which the spirit of my investigations is mainly dependent. To do this, I may at once state that all the great masters of Logic have observed that there are two points which must be rigorously attended to in correct systems of reasoning. First :—That the propositions employed as premises are unambiguous, and correctly understood. Second :—That the steps (the auxiliary operations and and Loci of Apollonius, &c. 25 theorems) by means of which the conclusion is drawn from those premises, are true, unambiguous, and correctly understood. This being borne in mind, it is evident that if from a given point in a given indefinite straight line, we were told to cut off apart equal in length toa given finite straight line, we should naturally ask in which direction from the point we are to take the part; as the problem would be ambiguous if either direction should not be answerable to the end in view. And if the solution of some other problem depended on this opera- tion (as just defined), and that one only of the parts which can be cut off is applicable, then it is evident there would exist an ambiguity in the solution. The method of indicating opposite directions on the same straight line in distinct terms—such as positive and negative directions, or right and left directions, obviates this difficulty ; but though long since adopted in Trigonometry and Algebraic Geometry, it is only in the modern French pure geometry, that it has been consistently introduced. Again, ifin any general investigation or construction it were necessary to draw a straight line through a particular point, making an angle of a given magnitude with another straight line, then, as two such lines can be drawn through the point, and that but one of them may be answerable, it is clear there should be a precise method of indicating each of these lines. Further, if on any straight line, for imstance, a particu- larised one of the last two, it werenecessary tocut off a segment from a point therein whose length should have some peculiar relation to other magnitudes and positions, and that but one segment from the point would be answerable ; then, too, it is obvious we should have a method of particularising directions in one straight line in respect to the directions in others. Yet itis only in my previous papers such methods are either advocated or applied. And without those improvements in the manner of indicat- ing angles, it is not only the elementary geometry of the straight line and circle that suffers, but also the conic sections and higher departments; for there, too, geometers have failed to expose the general truths comprehended in the theory. One instance of this is supplied by the following well-known theorem :—‘‘ When the base AB of a triangle is given in position and magnitude, and that the difference of the angles CAB, CBA at the base is constant, then will the locus of the vertex C be a hyperbola.” For, as the locus of the vertex is not a hyperbola under 26 The Three Sections, Tangencies, these conditions, but part of one, it follows that, in all inquiries in which the theorem is used, the results must be defective in generality. The complete theorem which should replace it is as follows :— If aa and 0b are fixed straight lmes through fixed points A and B, should ce and dd be any other pair of straight lines through A and B, making the angle cc right to aa equal to the angle 64 right to dd, then will the locus of the inter- section of cc and dd be a rectangular hyperbela through the fixed pomts A and B. Indeed, a due consideration of the requirements of a com- plete logic, or of the laws of nature, will show that the im- provements are necessary to the explicit enunciations of implied operations, and confer precision and generality on most important theorems; and are therefore, in so far as these are understood, a correct step in the advancement of pure science. Finally, I think it right to remark that, in mdicating opposite formations of magnitudes, I have purposely avoided the terms positive and negative, as so many meanings are given to these words by metaphysicians and others. Besides, the introduction of right and left renders the language more elegant, and often affords important advantages in allowing us to decide, according to circumstances, whether right or left should be indicated by plus or minus. Martin Garpiner, C.K. DEFINITIONS. 1. If astraight line, which we may conceive produced to in- finity in its primitive directions, be supposed to become rigid, and one point of it to be permanently fixed, the rigid line being otherwise capable of movement in any plane in which it may lie, then it is evident that there are but rwo ways of revolving the line im this plane; one being by means of a “vight”’ rotation, and the same as that in which the hands of a watch move if the dial-plate be towards us, in the plane; and the other being by means of the contrary, or “ left” ro- tation. 2. If AA and BB be two straight lines, and I their point of intersection, then the “angle [A right to B,’ means the angle formed at I by a rigid line having [ as a fixed pivot, and and Loci of Apollonius, &c. 27 revolving from a position in AA by a right rotation until its first arrival into the position BB, the revolving line being supposed produced indefinitely on both sides of the pivot. And a similar meaning applies to the term “angle IA left es i 3. If AA and BB be two straight lines, and I their point of intersection, then ‘‘ angle IA right round to IB” means the angle formed at I by a straight line having one of its ex- tremities in this point, revolved by right rotation from the actual direction IA until it arrives in the actual direction IB. And a similar meaning applies to the term “ angle LA left round to IB.” 4. If AA and BB be two straight lines, and I their point of intersection, the angle “‘right AB” means the angle IA right round to IB, and the angle “‘ deft 4B” means the angle TA left round to IB. 5. The angle (AB) means either one or the other of these last two, indifferently. NOTE. Tt is necessary to restrict the meaning of the term “ angle (AB),” given in Chasles’ Géométrie Supérieure, to that which has been just defined; for otherwise, his enunciated pro- perties of the homographic pencils will not hold good as to sign, See my paper entitled “ Improvements in Fundamental Ideas and Elementary Theorems of Geometry,” in the T’rans- actions for 1859. SECTION OF RATIO. Given the points P and R in the given straight lines MM and NN; through a given point A to draw a straight line CAD to cut the given lines in C and D, so that the segments 28 The Three Sections, Tangencies, PC and RD shail be to each other in the given ratio of m to n, ( = being of given magnitude, and of known sign in respect to directions on MM and NN). ANALYSIS. Suppose on MM and NN we take PQ and RS, so that PQ: RS:: PC: RD:: m: n, and that we draw PO and QO making the angles PO right to Q, and QO right to P re- spectively equal to the angles RA right to 8, and SA right to R. Then it is evident that the triangle POC is similar to RAD, and that the angle OC right to P is equal to the angle AD right to R. Hence, H being the point of intersection of PO and RA, it follows that a circle can pass through AOC and H; but A, O, and H are known points: therefore the point C, in which the circle AHO cuts MM, is known, and therefore also the lime CAD. Bat COMPOSITION. On MM and NN take segments PQ and RS, having to each other the given ratio of m:n; draw PO and QO, making the angles PO right to M, and QO right to M equal respectively to the angles RA right to N, and SA right to N; through A, O, and the intersection H of.PO and AR, de- scribe a circle; through either poimt C, in which this circle AHO cuts MM, draw CA to cut NN in D: then will CAD be an answerable line. For draw OC. The angle AH or AR right to C or D is equal angle OH or OP right to C, and therefore since the angle PO right to Q or C is equal angle RA right to S or D, it is evident that the triangles POC and RAD are similar, and that PC: : RD. :: PO . RAG PQ er Sr ine DISCUSSION. It is evident that when = is restricted as to sign, there is but one point O, one circle OAH, and two answerable points C (real or unreal). . If ~ be unrestricted in sign, then, obviously, there are two points O, two corresponding circles AOH, and, therefore, four answerable points C. Moreover, as the points O must and Loci of Apollonius, &c. 29 be on opposite sides of MM, it is evident two of these points C must be always real. . “ye a ae ° m Limiting Values for the Ratio —. When A and H are on opposite sides of MM, the corre- sponding points C are always real; but when A and Hare on the same side of MM, the reality of the points C is dependent on the position of O, or, which amounts to the same thing, on the value of ~ . Again, since PO: RA :: PQ: RS, it is evident that if O’ and O’ be the points in which the two circles through, A and H touching MM again cut PH, then will ;) and ix be the limiting values of “. Moreover, it is evident that, according as any value of “ is comprehended between these limits, or equal to one of them, or not compre- hended between them, so will the corresponding points C be imaginary, or real and coincident, or real and distinct. Porismatic Relations of the Data. It is evident the problem is indeterminate only when the circle AOH is indeterminate. When O coincides with A, and that PO cuts AH, then H also coincides with A, and the circle AOH is infinitely small; but when O coincides with A, and that PO and RA form one straight line, then it is obvious that any circle touching this straight line in A is an answerable circle AHO: therefore in this case there are in- numerable answerable points C and lines CAD. The problem under these last conditions (viz., when we have MM parallel to NN, and PRA a straight line, and the ratio = equal to fs) is said to be ‘porismatic’”’—any straight line CAD through A being an answerable line. Remarks. 1. When MM and NN are parallels, it is evident PO and RA are parallels and that H is at infinity (when O and A are not coimcident), and the circle AOH infinitely great. In this case the straight line AO, lying in the infinite circumference, will give one point C in its intersection with MM: the other point C is evidently at infinity on MM. 2. In all cases QO and SA intersect in the circumference of the circle AOH. | 30 The Three Sections, Tangencies, THE SECTION OF SPACE. M G \ Given the points Q and U in the gwen straight lines MM and NN; through a given point A to draw a straight line CAD cutting MM and NN in C and D, so that QC.UD shall be equal to m.n (where m.n is a given magnitude of known sign in respect to the directionson MM and NN). ANALYSIS. Suppose that in the given lines we take the segments QP and UR such that @P.UR = QC.UD = m.n, and that we draw PO and QO making the angles PO right to M, and QO right to M, respectively equal to AR right to U, and UR right to A. Then the similar triangles POQ, ARU, give @O.UA = QP.UR, and therefore @O.UA = QC.UD; and .°. as the angle QC right to O is equal the angle UA right to D, the triangles COQ and ADU are similar, and the angle CO right to Q is equal AD right to U; but FO right to Q is equal AR right to U; therefore it follows that angle AR right to D or C is equal OP right to C. Hence, if H be the point in which RA and PO intersect, a circle can pass through AOH and C: but the points AOH are known; therefore the point C in which the circle AOH cuts MM is known, and hence CAD. \ COMPOSITION. In the given lines MM and NN take any two segments QP and UR, such that @P.UR = m.n; draw PO and QO, mak- ing the angles PO right to Q, and @O right to P equal re- spectively to AR right to U,and UR mght to A; through A, and Loci of Apollonius, &e. 51 O, and the intersection H of RA and PO, describe a cirele ; through either point C in which this circle cuts MM draw CA to cut NN in D: then will CAD be an answerabie line. For draw CO. The angle OH or OP right to C is = AH or AR right to C or D; and the angle PO right to Q = AR right to U; therefore the angle CO right to Q is equal AD right to U; hence the triangles ADU, COQ are similar, and QC.UD = QO.UA. But the similar triangles PQO, AUR give Q@O.UA = QP.UR;; therefore QC.UD = QP.UR = aun. DISCUSSION. When 7.” is restricted in sign (as im the enunciation), there is evidently but one answerable point O, and therefore but one circle AOH, and two points C, real or unreal, according as the circle AOH cuts MM im real or imaginary points, If m.n be unrestricted as to sign, then there are evidently two answerable points O, and therefore two circles AOH, and four points C, and lines CAD : moreover, it is evident that the points O are on opposite sides of MM, and therefore that two of the points C must be always real. Limiting Values for m.n. It is evident the poimts C can be imaginary only whe A, O, and H are on the same side of MM. We know one poimt A in the circle AOHC, but, in order to arrive in a sim- ple manner at the limiting positions for the circle AOH, it would be well if we could find another point in the circum- ference. We can find sucha point. For, if T be the point in which QO again cuts the circle, the angle AH right to T = OH right to T, and is therefore = RA or RH right to N; and hence AT is parallel to NN, and the point T in which it cuts QO is known. Now it is evident that by putting O’ and QO’ for the points in which the circles through A and T, touching Mii, cut QO, then will UA.QO’ and UA.QO’ be the required limits. Moreover, it is evident that according as any value of m.n is comprehended between these limits, or equal to one of them, or not comprehended between them, so will the correspond- ing points C be imaginary, or real and coincident, or real and distinct. Porismatic Relations of the Data. Itis evident the problem becomes indeterminate only when the circle AOH becomes indeterminate. Now the circle 32 The Three Sections, Tangencies, AOH evidently becomes indeterminate when O and A coin- cide, and that QOAT is parallel to NN. In this state of the data all pomts in MM are answerable points C, and the problem is said to be “ porismatic.” It is evident that UA is parallel to MM when QOA is to NN, and that m.n (in this porismatic case) is equal QA.UA. THE DETERMINATE SECTION. Given two pair of points P,S, and Q,R, ina straight line MN ; to find a point C in the line such that PC .SC : QC . RC : 0: k.(in which the sign of < is known, as well as its magnitude ). ANALYSIS. Suppose we assume a point A, and that we draw PG and QG making the angles PG right to Q, and QG right to P respectively equal to RA right to §, and SA right to R; and that E and F are the poimts ia which AC cuts QG and PG. The triangles CPE, CQF, are similar to CRA, CSA, and we evidently have PE . SA: QF .RA:: PC .SC: QC. RC :: 2: k; and therefore PE has to QF the known ratio of RA.J to SA.k. Hence (see Porism 4 in Transactions for 1859), the circle EFG passes through a known point O in the circumference of circle PGQ, which is such that PO : QOe ster ey OPE. Again, the angle EF or EC right to O = GF or GQ right to O = PQ or PC right to O; therefore a circle can pass through EPC and O; and hence, as AR is parallel to PE, if H be the point of intersection of PO and AR, it follows that a circle can pass through OHA and C; but O, H, and A are known points; therefore the circle OHA is known, and also the point C in which it cuts MN. COMPOSITION. Assume a point A (not in the given line); draw PG and QG making the angles PG and QGright to M equal re- and Loci of Apollonius, &c. 30 spectively to the angles RA and SA right to N; describe the circle PGQ, and in it find O (on the same or opposite sides of PQ with G according as OG and ERA have like or un- hike signs) such that PO : QO:: 1.RA : k&.SA; draw OP to cut AR in H; describe the circle OAH: either point C in which it cuts MN is an answerable point. Let E and F be the points in which GP and GQ cut AC. It is evident PE is parallel RA, and QF to SA, and therefore that PE.SA : QF.RA :: PC.SC: QC.RC. Again, the angle EP right to C being equal AR or AH right to C, it is equal OH or OP right to C; therefore a circle can pass through OCPE, and the angle EC right to O = PC or PQ right toO = GQright to O; hence, acircle can pass through GFOH, and therefore (see Porism 4, Transactions for 1859), PE : QF :: PO : QO::J.RA : &.SA, and therefore PE.SA : QF.RA :: 7 : k, and consequently PC.SC : OG BUi:: ¢.:k, DISCUSSION. It is evident the point of intersection I of OQ and SA isin the circumference OHA (for the angle IQ or IO right to A = QO right to G = PO right to G, and ... = HO right to A). When ; is (as is supposed in the enunciation) confined to a particular sign, there is but one point O, one circle AHO, and therefore two (and but two) answerable points P—both real or both imaginary. But if i were unrestricted in sign, it is evident there would be two points O, and therefore two circles OHA, and four points C. Moreover, since the points O must be on different sides of MN, two of these points C must he always real. Limiting Values for the Ratio : When the segments PS and QR lie partly on each other, the points H and I lie on opposite sides of MN, and therefore the corresponding point C must be always real. When one of the segments PS and QR lies entirely on the other, it is evident the points A and G are on the same side of MN, and therefore it is only when the ratio ; is positive, that O, H and Acan be on the same side of MN; in otherwords, the points C are real for all real negative values of - ; but for positive values of $ the points C are real only when the D 34 The Three Sections, Tangencies, eircle OHA cuts MN in real points. Hence, it is necessary to define the positive limiting values of £ so as to be enabled to know, @ priori, when the corresponding points C are real, &e. This is done by drawing the two straight lines PO'H’ to cut the circle PGQ and line AR in O’ and H’ such that the two circles AO’H’ shall touch MN (these circles can be easily described, since the point X, in which AG cuts the circle PGQ, is common to all circles AHO. For the angle XG right to O = PGright to O = HA right toO = XA right to O); for aD er ia SAO! RA.PO' , - then the limits are GoGo and sq:qo'; and these lmmits are evidently such that according as any value of < lies outside them, or is equal to one of them, or is comprehended between them, so will the corresponding points C be real and distinct, real and coincident, or imaginary. When the segments PS and QR have no part in common, it is evident A and G are on opposite sides of MN, and that it is only when the given ratio is negative that the points OHTA can be on the same side of MN, and .-. only that the points C can be imaginary. Hence, in this case, it is neces- sary to define the negative limiting values of =. These pee iN RA.Po' RA.Po! : : limiting values §iQ,” SaA-Qd are obviously found in the same manner as in last case, and like remarks as to their nature apply. When Q coincides with §, it is evident the points C are always real, and that one of them is coincident with QS. When P coincides with R, it is evident the pomts C are real, and one of them in PR. When RB and S are coincident, and P and Q are distinct, then AR and AS are coincident, and G is at infinity, and .-. O is in the straight line PQ; moreover, H coimcides with RS; .. the points C are real, and one of them coincident with R. When P and Q are coincident, and R and § distinct ; then G coincides with P and O; but although the triangle POQ is infinitely small, it is known in species, and therefore POH is known in position, and hence the circle OHA. The points C are real, and one of them coincident with PQ. Porismatic Relations of the Data. If R and § be coincident, and that we have P and Q also coincident, then as we may have conceived PQ and RS to and Loci of Apollonius, &c. 35 have had any peculiar relation, such as a constant ratio, &c., during their diminution, it is evident we may suppose G any- where whatever in the line PQG. And it is clear that for all values of £ other than unity, the pomt O must coincide with PQ, and one point C be coincident with PQO, and the other point C with RSH. But for the value of + = unity, the point O may be anywhere in the circle GQP (which touches MM in PQ), and .*. a pomt C may be anywhere in MM— the state of the data being “ porismatic.” And it is evident that when P coincides with R and S with Q, and that - = unity, then, also, will the problem be “ porismatic.”’ Peculiar Case. If in MN we suppose SR and RU equals respectively to / and k; then we have PC.SC : QC.RC :: SR: RU. Now, if we suppose R and U to remain fixed, and S to become greater and greater in distance from the fixed points, until it vanishes at infinity, then for points C at a finite distance, we shall have SR = SC, and hence PC.RU = OC.HE. This case of the problem can be expressed as follows :— “ Given three points, P,Q, R, ina straight line, to find another C in the same, such that K being a line of given length (and known sign in respect to the directions on MN), we shall have PC.K = QC.RC.” - he solution may evidently be worded thus:—In MN make RU = K; draw RA (not in MN) equal RU; draw PO and QO, making the angles PO right to Q, and QU right to P equal respectively to AR right to U, and UR right to A ; produce PO to cut RA in I; describe the circle OHA, and it will cut MN in the required points C, C. 36 The Three Sections, Tangencies, GENERATING PROBLEM TO THE THREE SECTIONS. M Given the points P Q on « straight line MM, and the points R S ona straight line NN; through two given poinis A B to draw two straight lines AI, BI, making the angle IA right to B of a given angular magnitude 0 right, and such that C and D being the points in which AI and BI cut MM and NN, we shall have PC.SD : QC.RD :: 7: k; i being of given mag- nitude and known sign in respect to the directions on MM and NN). ANALYSIS. The circle AIB is evidently known. Suppose we draw a straight line CHI’, making the angle CH or CF right to P or Q equal the angle DB right to R or §, and that through P and Q we draw PE and QF to cut it, so that the angle PE right to C = RB right to D, and the angle QI right to C = SBright to D. Then the triangles CPE, CQF are similar to the triangles DRB, DSB; moreover it is evident that from these triangles we have the relation PE.SB : QF.RB:: POSD {OC RD ib ep. Let G be the point of intersection of PE and QI’. From the last proportion we have PE to QF in the known ratio of 1.RB to k.SB; aud therefore (see Porism 4th, in Transactions of 1859) the circles EGI', CQF, CPE pass through a point O in the circumference of the circle PGQ, which is such that PO : QO :: 1.RB : %.5B; and hence, as PG and QG are known, the point O in the circle PQG is known. Again, the angle OC right to P = EC right to P= BD and Loci of Apollonius, &c. 37 right to R; and therefore if L be the other point in which RB cuts the circle AIB, and H that in which OP cuts AL, we have the angle OC right to P or H = BD or BI right to R or L = AI or AC right to L or H; hencea circle can pass through O, H, A, and G; but O, H, and A are known points ; therefore the point Cin which the circle OHA cuts MM is known, and therefore the point I in which CA cuts the circle AIB, and also the point D in which IB cuts NN, COMPOSITION. Through A and B describe the circle AIB, such that I being any point in its circumference, the angle IA right to B = Oright; draw PG and QG, making the angle PG right to M = RB right to N, and the angle QGright to M = SB right to N ; describe the circle PGQ, and 7 - (on the same or opposite side of PQ with G, according as aa & and ESB have like or unlike signs) find O such that PO : QO :: LRB : k.SB; draw RB to cut the circle AIB in L; draw OP to cut Al in H; describe the circle OHA ; through either point C in which the circle OHA cuts MM draw CA to cut the circle ABLin I; draw BI to cut NN in D: then will AI and BI be as required. For through the other point E in which the circle OCP euts GP, draw CE to cut QG in F. The angle KC right to P is = OC right to P or H = AC or Al right to H or L = BI or BD right to L or R; there- fore, since the angle PC right to E — RD right to B, the triangles PCE, RDB are similar, and CP or CQ right to E or F is equal DR or DS right to B. And since the angle QG or QF right to P or C is equal angle SB right to R or D, therefore the triangles CQF, DSB are similar. Now from these two pair of triangles we evidently have PE.SB : QF.RB :: PC.SD : QC.RD; but (see Porism in Transactions for 1859) we have PE : QF :: PO : QO:: L.RB : k&.SB, and therefore PE.SB : QF.RB eee nenee PC.SD :'QC.RD'?? 0 2k. DISCUSSION, Since there are two points of intersection C, there are two solutions to the problem, both real or both imaginary, accord- ing as these points are real or imaginary, If the ratio 4 be unrestricted as to sign, then it is evident 38 The Three Sections, Tangencies, there are two answerable points O, and therefore two circles OHA, and four points C. Moreover, the points O being necessarily on opposite sides of PQ, it follows that two of these four points C must be always real. Limiting Values for 6 Right. * There are evidently limits to 0 right only when A and O are on the same side of MM. . Now the point O is found independent of the magnitude of 6; and it is evident that by describing the two circles through A and O which touch MM, and putting H’ and H’ for the points in which OP cuts them again, and I’ and I’ for the points in which AH’ and AH’ cut RB, then will the angles VA right to B and I’A right to B be the limiting values for the angles 6 right. Andif X be the pomt im which AO cuts I’I', it is evident that according as a and = have like or unlike signs, so will any straight lme through A cutting I'l’ and H’H' in L and H give = and #4 of like or unlike sign. Hence it follows that when or and = have like signs, the limiting values include between them all values (and no others) of the angle 6 right, for which AI and BI are imaginary ; and when our and be have unlike signs, the limiting values have outside them all values (and no others) of angle 6 right, for which AI and BI are imaginary. And when 6 right is equal cither of the limiting values, the lines AI are coincident and real. Limiting Values for the Ratio = When the points C may be imaginary, the point O must evidently fall on the same side of PQ with the points A and H. And it is evident we arrive at the limiting values of i, or GORE» or (which is equivalent) of ae by finding the points o’ and o’ in the circle PGQ, so that h’ and h’ being the points im which Po’ and Po’ cut Al, the circles Ah’o’ and Ah’o! touch MM. — Moreover, it is evident that the two ratios eae and RG are the required limits, and that they have the same sign, and that according as any value of 4 (having like sign) is of a * Geometers who do not adopt my improved methods of indicating angles, will find it impossible to define the Zimits of the angular magnitude @, and Loci of Apollonius, &c. 39 magnitude comprehended between them, or equal to one of them, or not comprehended between them, so accordingly will the circle AOH corresponding cut MM in two imaginary points, in two real and coincident points, or two real and dis- tinct points C. Porismatic Relations of Data. 1. It is evident the problem becomes indeterminate whenthe circle OAH becomes indeterminate, &c. Now, if the circle PQG passes through A, and that the point O coincides with A, then H on AL will also coincide with A; if, AL does not coincide with OP; and, as the chord OH on OP is equal zero, the circle AHO must be infinitely small. But if AL and OP coincide, then, although AH and OH are “infinitely smalls,”’ they lie both on AL, and therefore it is evident that any circle touching AL in A is an answerable circle AHO. ‘Therefore in this last state of the data the poimts C are innumerable, and the problem is “ porismatic,” as well as if we conceived O to move to A having OP parallel AL, and thus causing H to be indeterminate when O coin- cides with A. 2. If R coincides with §, and that we suppose Q to approach P until it comes to coincide with it, then G is at infinity, and the straight lime MM lies in the infinite circumference PQG. And for all values of + other than + the point O coincides with PQ, and one point C is coincident with PQ, and the other point C with the point in which AL cuts MM. But when - = |, then as the point O, and the point C coincident with O, may be anywhere in MM, the problem is said to be “ porismatic.” Remarks concerning Particular Cases. 1. Since PC.SD : QC.RD :: 1: &, or, which is the same thing, since PC.(RD - RS) : (PC —- PQ).RD :: /: &, therefore, when 1 = k, we have PC.RS = PQ.RD or PC : RD:: PQ RS Hence we derive a method of solving the problem—* Given the points P, R in the given straight lines MM, NN; through two given points A and B to draw Al and BI making the angle 1A right to Bof a given magnitude 0 right, and such that 40 The Three Sections, Tangencies, C and D being the points in which AI and BI cut MM and NN, we shall have PC to RD wa a given ratio of m to n.” It is evident that the points G and O are coincident, and .. the solution of this enunciated problem can be worded as follows :— Take Qand S on MM and NN, so that PQ: RS:: m:n; through A and B describe the circle AIB, which is such that I being any point in it, we have the angle IA right to B = @right; throvgh P and Q draw PG and QG, mak- ing the angles PG and QG right to M, equals respectively to the angles RB and SB right to N, through L where RB again cuts circle ABI, draw AL to cut PG in H; describe the circle AHG ; through either pomt C in which it cuts MM draw CA to cut circle AIB again in I; a ae to cut NN in D; then will PC : RD : far: RS : n. 2. If we suppose not only J = k, but also 6 = vee and B coincident with A; then it is ev ident the problem becomes the ‘ Section of Ratio” of Apollonius. Moreover, it is evident the preceding solution to the Apollonian problem flows directly from the present more general problem, for in this particular state of the data we evidently have G coinci- dent with O, and B, L, and I coincident with A, and ARH in straight line, &c. 83. Since PC.SD : QC.RD :: 1 : k, if we suppose in NN SR and RU always equal / and k, we have PC.SD : QC.RD :: SR: RU Now, if we suppose R and U to remain fixed, and that S becomes infinitely distant, then for a point D at a finite distance, we have SR = SD, and therefore PC.RU = QC.RD. Hence (QC—@QP). RU = QC.RD, and (RU—RD) QC = QPS DUOC — PiU: Or, UD.QC = QP.UR = a known magnitude. Hence, we derive a method of solving the problem. “ Given the points U and Q in given straight lines NN and MM ; through two given points A and B to draw two straight lines AI and BI making the angle LA right to B equal a gwen angular magnitude @ right, and such that C and D bemg the respective points in which AI and BI cut MM and NN, we shall have UD.QC = mn.” Where m.n is given in sign, ‘and the directions on the given lines particularised. It is evident that in this case G coincides with P, and that circle GPQ touches PG at this double point, and .-. that the triangle QOP is similar to URB, &c. and Loci of Apollonius, &e. A Hence, the solution of this enunciated problem may be made as follows:—In MM and NN take P and R such that QP.UR = m.n; describe the circle AIB such that I being any point in it, the angle IA right to B = @ right; through P draw a line PG making the angle PG night to M = RB right to N, and describe the circle QOP through Q which touches this lime at P; find the point O in this cirele such that PO : QO :: RB: RU; through the point L in which RB again cuts the circle AIB, draw AL to cut POin H; describe the cirele AOH; through either pomt C in which this circle cuts MM, draw CA to cut the circle AIB again in I; draw BI to cut MN in D. Then will AI and BI be answerable lines. Moreover, it is evident that the angles PO and QO right to M are respectively equals to the angles BR and UR right to U and B, and that we can, therefore, determine PO and QO without drawing PG or describing the circle QPOG. 4. If in addition to the conditions of this third case, we suppose the angle 6 = zero, and B coincident with A ; then, it is. evident, the problem becomes the “‘ Section of Space” of Apollonius. Here again it is evident RAH is a straight line, and that the solution which I have given to this Apol- lonian problem has been derived from the present more general problem. 5. We have PC.SD : QC.RD :: 2: k, when the angle 0 has any finite magnitude; and it is evident that when 6 = zero, and that MM coincides with NN, we have PC.SC : QC.RC :: 1: k, which is the ‘‘ Determinate Section” of Apollonius. It is further evident that the solution just given to this Apollonian problem has been derived from the present more general one. REMARKS. If K be the other point in which SB cuts the circle AIB ; then it is obvious AK and QO intersect in a point T in the circle AHO. If U be the other point of intersection of the circle OGPQ with any circle AOH, and that PV be drawn parallel to AL to cut circle OGPQ in V, then will VUA be in one straight line. Therefore, as the point U is known independent of the ratio, the limiting circles Av’o’U passing through A and U can be hence easily described. It is also evident QV is parallel to AK, &e. 42 The Three Sections, Tangencies, Again it may be remarked that if we could solve the generating problem by a different method, we could thence derive other analogous solutions to the ‘the Three Problems of Section.’ However, instead of giving another solution to the problem, in which it is required to have PC.SD : QC.RD::J/: k. I will now solve the more extended generating problem in which it is required to have Cg PC.SD+p.s : QC.RD+ 9.7 :: 1: k, where the magnitudes and signs of the rectangles p.s, and g.7 are given. ANALYSIS. (The figure to be supplied by the reader.) Suppose we draw PG, QG, making the angle PG and QG right to M, respectively equals to SN and RN right to B.. Then the point G is given. If we draw CE and CF meeting PG and QG in E and F, so that the angles EC and FC right to P and Q shall be each equal to angle DB right to Ror 8. Then it is evident the triangles QCF, PCH, are similar to the triangles RBD, SBD, and that a circle can pass through CFEG. Moreover, it is evident PC.SD and QC.RD are respectively equal to PE.SB and QF.RB, and therefore we have PE.SB+p.s : QF.RB+ Garin ts k. And if in PE and QF we take the points J and T, such that JP.SB and TQ.RB are equals respectively to p.s and PP; then it is evident the points J and T are known, and that JE.SB : TE.RB :: 72: k. Now, from the porisms in Trans- actions for 1859, we know that the circle HFG will cut the oe circle JTG ina point O such that JO.SB : TO.RB :: 2: k; and therefore O is a known point im cir cle JTG. Again, let V be the point in which CA cuts circle CEFG, and U that in which GV cuts circle JTG, and H that in which UA again cuts circle JTGU. We have angle HO right to U or A = GO right to U or V=CO right to V or A, and therefore a circle can pass through COH and A; but the angle VG right to C being equal EG right to C it is equal DN right to B, and .°. if W be the point in which GV cuts NN, a circle can pass through VIDW, and the angle WV right to D or N is equal the angle IV or IA right to D or B, and thererefore GW is known in position. Moreover, the pomt U where GW cuts circle and Loci of Apollonius, &c. 43 JTG is known, and the other point H in which UA cuts eircle JTGU, and .*. the point C in which the circle HAO euts MM, and hence CAI and BID. The composition, &c., may be easily made. However, in order to familiarise the methods of arriving at the limits of angular magnitudes in other questions, I will indicate the nature of the limits of @ right. Limiting Values for 6 Right. As the points O and A are known independently of 6, .*. it is evident that by describing the two circles through O and A which touch MM, and putting H’ and H’ for the pomts in which they again cut circle JTG, and U’ and U’ for those in which AH’ and AH’ again cut this same circle, and W’ and W’ for the points in which GU’ and GU’ cut NN, then will the angles W’G right to N and W’Gright to N be the limiting values of 6 right. And, if ’ and w’ be the pomts in which GA cuts the circle JTG and lne NN, it is evident that G may be regarded as a position of U corresponding to w’. Moreover, it is evident that if w’ lies between W’ and W’, and that h' is not inside or outside doth circles through A and O touching MM, then will the circle OAV’ cut MM in imagi- nary poits; and the hmiting values for @ right are evidently such as to include between them all values of 6 right (and no others) for which the lines AI and BI areimaginary. But if w' lies between W’ and W’, and that /’ is mside or outside both circles through A and O touching MM, then will the limiting values of @ right be such as to have outside them all values of @ right (and no others) for which the lmes AI and BI are imaginary. And it is further evident, that if w’ lies outside W’W’ and that 4 is not mside or outside both circles through A and O touching MM, then will the limiting values of @ right be such as to have outside them all values of @ right (and no others) for which AI and BI are imaginary; but if w’ lies outside W’'W’, and that /’ is inside or outside both circles through A and O touching MM, then will the limiting values of @ right be such as to include between them all values of 6 right (and no others) for which AI and BI are imaginary. When @ right is equal either limit, the lines AI are coinci- dent and real. Moreover, it is evident that when A and O are not on the same side of MM there are no real limits to angle 6 right, &e. 4A, The Three Sections, Tangencies, NOTES. 1. Itis evident that the porisms in the Transactions for 1859 can, with many others, be derived from the porismatic states of the data of this or the first Generating Problem of the Three Sections. 2. And very probably, the Greek geometers derived from this and other kindred problems, by means of projections, &c., part of their ‘porismatic knowledge’ which is now known as the ‘ anharmonic properties of pencils and divisions.’ 3. In the investigations of the limits of the angles 6 right in problems 1 and 2 of my paper in the Transactions for 1859, it would be well to omit all the words from ‘ And it is more- over evident,’ &c., and substitute the followmg :—And when the circles iBC circumscribe the portion of circle ACH which is not within or outside both the circles BA?, the lmitmng values include between them all values of 6 right, and no others, for which CO and BO are imaginary; but when the circles iBC circumscribe the portion of circle ACH which is within or outside both circles BiA, then the limiting values have outside them all values of 6 right, and no others, for which CO and BO are imaginary. At the limits the lies CO are real and comcident. THE TANGENCIES. FIRST SOLUTION. (See Plate.) To describe a circle to touch three given circies A, B, C. ANALYSIS. Let D, E, F, be the respective pomts of contact of the required circle with the circles A, B,C. Then DE passes through O a known centre of similitude of circles A, B; and DF passes through P, a known centre of similitude of the circles A, C. Now if D’ be any assumed poimt in circumference A, and that E’, F’, are the dissimilar pomts in which OD’ and PD’ cut circumferences B and C; then PD’.PF’ — PD.PF, and OD’.OL’ = OD.OE; and it follows that the circles D’E’F’, DEF, have PO as radical axis. Let d’, e', and f’, be the other pomts in which the known circle D'E’F’ cuts the circles A, B, C. It is evident the THE TANGENCIES, First Solution. es CT and Loci of Apollonius, &c. 45 straight line D’d' and the common tangent to the circles A and DEF at D, cut each other in PO the radical axis of the circles DEF and D‘E’F’. And for like reasons it is also evident that the intersection of E’e’ and the tangent to circle B at E, and also the intersection of If’ and tangent to circle C at F, are in PO. But the intersections of the straight limes D’d’, He’, and If’ with PO are known; .’. the tangents from these points to the respective circles A, B, C, are known; hence, the poimts of contact D, E, F, being known, the circle DEF is known. Or, having found either point of contact the others can be easily determined. ‘Thus for mstance when D is found, then and F are the dissimilar pomts m which OD and PD cut the circles B and C. COMPOSITION. Find O a centre of similitude of circles A and B; find Pa centre of similitude of circles A and C; through D’ any assumed point in circumference of circle A, draw OD’ and PD’ to cut the circumferences of B and C in the points Ii’ and ¥’ dissimilar to D’ on circumference A; describe the circle D’E'F’ and draw D‘d’, ii’e’, If’, its respective chords of inter- section with the circles A, B, C, to cut the straight lime PO in a, 6, c.; draw aD tangent to the circle A; draw OD and PD to cut the circles B and C m the poimts H and F dissimilar to point D on circle A; describe the circle DEF. Then will _DEF be a required circle. For OD.OE being = OD’.OE’, and PD.PF = PD’.PF’, it follows that OP is the radical axis of the circles DEF and D’E’t’, and therefore that aD is tangent to the circle DEF as well as to circle A at the pomt D: and hence the circle DEF touches circle A at D. And since ODE passes through the peimt of contact D of the circles DEF and A, and that O is a centre of similitude of circles A and B, and that the points D and E on circles A and B are dissimilar, .*. the circle DEF touches the circle B in EK. And for similar reasons the circle DEF touches circle C in F. NOTES. It is well to observe that we can find the poimts D and F from E (E being the point of contact of a tangent from 6 to 46 The Three Sections, Tangencies, circle B) by drawing OE to cut the circle A in the point D dissimilar to the point E on circle B, and then PD to cut the circle C in F the point dissimilar to D on circle A. And in like manner we may find D and E from F the point of contact of the tangent from e¢ to circle C. Moreover, it is evident that we can find D even when the circle A is infinitely great, for the tangents from a to circles A and D’‘E’Y”’ are equal. As there are two points O, and two poimts P, there are four lines PO and .:. evidently four answerable points a; and hence, as there are two tangents from each point a@ to the circle A, it follows that there are in all eight answerable circles DEF, real or imaginary in pairs. Now DD, EE, and FF, are the polar chords of the circles A, B, and C im respect to the points a, b,c im PO. Let Q be the centre of similitude of the circles B and C through which EF passes. Then OPQ is straight. And as DD must pass through a’ the pole of OPQ in respect to circle A, and that EE must pass through 0’ the pole of OPQ in respect to circle B, and that FF must pass through c’ the pole of OPQ in respect to circle C ; hence it follows, because ODE, PDF and QFE are straight lmes, that DD, EE and FF must meet in the radical centre R of the given circles A, B, C. And this indicates the method of solution given by Gergonne. It is also easy to see that DD passes through A’ the extremity of the diameter AA’ of the known circle D'Ad'; and EE ev idently passes through B’ the extremity of the diameter BB’ of circle BE’e’ ; and FF passes through C’ the extremity of the diameter Gc of the circle CF 7’ ~—Hence other methods of solution. We may also remark, that as there are four points a, and that the four polars of these points in respect to the cirele A pass through the radical centre R of the three given circles, it follows that the four points @ are situated in the polar of R in respect to the circle A. And, similarly, the four points 6 are on the polar of R in respect to the circle B, and the four pomts ¢ are on the polar of R in respect to circle C. SECOND SOLUTION, (See Plate.) To describe a circle to touch three given circles A. B, C. THE TANGENCIES Solution. Second and Loci of Apollonius, &c. 47 ANALYSIS. Let O be the centre of the required circle, and let D, E, F be its points of contact with the given circles A, B, C. Then OAD, OBE, OCF are straight lines. Since OD = OH, if OG bisects the angle OD right round to OB, and that AG is perpendicular from A on OG, then is the locus of G a known circle having its centre in the middle point of AB, and such that its diameter MP intercepted by AG and OG is parallel to AO. Similarly, smce OD = OF, it follows that if OH bisects the angle OD right round to OF, and that AH is perpendicular from A on OH, then will the locus of H be a known circle having its centre equally distant from A and C, and such that the diameter NQ intercepted between AH and OH shall be parallel to AO. And if we assume any auxiliary circle having A as centre, and that AH.AN’ is equal to the square of its radius, then the locus of N’ is a known circle N‘H’Q’ such that H’ being the other point in which AH cuts it, the segments cut off by the chords NH and N’‘H’ are similar. And, for like reasons, ifon AG we take the point M’ such that AG.AM’ = the square of the radius of the auxiliary circle, then will the locus of M’ be a circie having with MGP the poimt A as a centre of similitude, and such that, if G’ be the other pomt im which AG cuts it, the segments cut off from it and circle PGM by the chords MG and M’G’ are similar. Let I be the point in which AO cuts M’N’. Since the angles G and H are right, and that AG.AM’ = AH.AN’ = square of radius of auxiliary circle, it follows that M'N’ is perpendicular to AO, and that AO.AI is equal the square of radius of auxiliary circle. Again, the angle M’N’ right to A being equal to the angle OA right to G = PM right to G, it is ... = P’M’ right to G’, and .*. M’N’ touches circle M’G’P’ in M’. Similarly, since angle N’A right to M’ = OH right to A = QH right to N, it is .". equal Q’'H’ right to N’, and .:. M’N’ touches the circle N’‘H'Q’ at N’. Hence as M’N’ is a common tangent to two known circles it is itself known, and .°. the centre O which is the pole of M'N’ in respect to the auxiliary circle, is known, and .’. also the circle DEF. And since the radius of the circle PGM can be taken equal to the half sum or half difference of the radii of circles A, B, and that the radius of circle NHQ can be taken equal either 48 The Three Sections, Tangencies, the half sum or half difference of the circles A, C, and that to each of the resulting pairs of circles there are two answerable tangents, .*. it is evident there are four pair of answerable centres O, and .*. eight solutions to the question which are real or unreal in pairs. The composition may be easily made. And it may be as well to remark that when we suppose the circle C infinitely great, then will the circle NHQ also be infinitely great ;. and its infinite circumference bisects all straight lines drawn from the pomt A to the infinite circumference of circle C, &c., &e. It may be right to observe that by introducing an auxiliary curcle into the Fourth Solution in a similar manner to that in this solution, we can make it intelligibly applicable to the minor cases which now escape it ; but though this might be an advantage as regards the greater generality obtamed, it would not indicate such neat solutions to the leading cases. THIRD SOLUTION. (See Plate.) To describe a circle to touch three given circles A, B, C. ANALYSIS. Let D, E and F be its pomts of contact with the given circles A, B and C. Then DE, DF and Ei pass through the respective points O, P and Q, centres of siumilitude of the given circles which are in one straight line. Let Gand L be the other pomts in which DE cuts the circles A and B; and let H and M be those in which DF cuts the circles A and C; and let 8 be the pomt m which GH cuts the axis of similitude OPQ. Then OS has to OQ the known ratio which OG has to OH, and .°. the point 8 is known. Now the ratio of OG.DH to PH.DG, which is the same as that of OS to PS, is known; and the ratio of PD.PH to OD.OGisalso known ; .:. the ratio compounded of these ratios or that of PD.DH to OD.DG is known: and hence as DH : DG: : DE: DEH, it follows that the ratio of PD.DF to OD.DE is known. Or—which amounts to the same—the ratio of OH.FD to PF.ED being the same as that of OQ to PQ is known; and the ratio of PF.PD to OH.OD is (the same with that of THE TANGENCIES Third Solution. and Loci of Apollonius, &c. 49 PF.PM. rad B to OE.OL. rad C) known; hence the ratio of PD.DF to OD.DE is known. Let N be the point in which a tangent to the circle C at M euts the straight hne ODE. It is evident DE.DN = DF.DM, and that DE.DN has to OD.DE a ratio compounded of the known ratios of PD.DF to OD.DE, and of DM to PD. Or—which amounts to the same—it is evident DE.DN has to OD.DE a ratio compounded of the known ratios of OQ to PQ, of PD.PF to OD.OE, and of DM to PD, which may evidently be expressed as the ratio compounded of the ratios of OQ to PQ and of PF.DM to OD.OE. Hence it follows that the point N must be in the circum- ference of a known circle NXX haying with circle A the pomt O as centre of similitude. Moreover, if K be the other pomt m which DE cuts circle NXX we have the angle XK right to N = angle FE right to D, and .*. = angle ND right to M; and hence MN is tangent to the circle NXX at N. Now MN being a common tangent to two known circles, it is itself known ; and .*. the other pomt F in which the straight line PM cuts tavele C is known, as also the poimt D in which it cuts the circle A similarly to the pot M on circle C; and the point E in which ON cuts the cir cle B similarly to poimt N on circle NXX is known: and .-. the required circle DEF is known. COMPOSITION. Find O, a centre of similitude of A and B; find P a centre of similitude of A and ©; and Q a centre of similitude of B and C in the line OP; and find the pomt S in OPQ such that OS : OQ :: rad A : rad B. Through O draw a straight lime OD‘E’ to cut the circles A and B in dissimilar points D’ and E’; draw PD’ to cut circle C in the point F’ which is dissimilar to D’ on circle A, and to eut it again in M’; find the pomt N’ in OD’‘F’ such that D‘N’.D’E’ shall have to O’D’.D’E’ the ratio compounded of the ratios of D’M' to PD’ of OS to PS and of PD.PH to OD.OG, —or which amounts to the same—such that D’N’.D’E’ shall have to OD’.D’E’ a ratio compounded of the ratios of OQ to PQ and of PE’.D'M’ to OD’.OE’; draw N’ I parallel D’A to cut OAB in I; with I as centre and radius IN’ describe a circle ; draw MN a common tangent to the circles C and I; E 50 The Three Sections, Tangencies, through the point of contact M, on circle C, draw PM to cut the circle C again in F, and to cut the circle A in D similarly to the pomt M on circle C; draw ON to cut circle Bm E similarly to the point N on circle I; describe the circle DEF. Then is DEF a required circle. NOTES. This method of solution holds intelligibly good in all cases in which neither of the circles A or C is infinitely great. When the circles A and B are injinitely smalls the centre of similitude O may have any position whatever in the line AB, as the ratio of their radii may be of any magnitude; and similar remarks apply to the centre of similitude Q when the circles B and C are infinitely smalls. By fixing the ratio of these infinitely small circles, we fix the positions of the centres of similitude ; and it is evident we may suppose one of them infinitely small in respect to the other, so as to have the centre of similitude comcident with this other in respect to finite distances. And similar remarks apply as to the ratios of infinitely great radii. This solution furnishes three methods to the case in which two of the given circles are finite and the third infinitely small. And that one in which we have A the infinitely small circle is in substance the same as what is given by Monsieur Auguste Cauchy. ; We are furnished with two methods for the case in which two of the circles are infinitely smalls, and the third finite :— one of which (when A and B are the infinitely smalls), 1s in substance the same as what is given by Pappus as the solution to this case from the Work of Apollonius. However, here as elsewhere, when I speak of a general solution being inapplicable to any case or cases, it is to be considered inapplicable only in a graphical point of view, for a general solution holds mentally good in all cases, even when quantities may be infinitely great or small; and the mind’s conviction in such cases is established by its knowledge concerning properties of finite quantities and its own power of legitimately applying the principle of ‘ continutty’ derived, in degree, from this knowledge. It is also to be observed that owing to our imperfect know- ledge of infinitesimal geometry, or to the nature even of this geometry, it may often happen that we cannot intelligibly arrive at some necessary theorem from one point of view, so " es aie bi MS isis). Hp POP vats.” < scribe asad bejectyy) THE TANGENCIES Fourth Solftion ao CK) and Loci of Apollonius, &c. 51 as to employ the steps in a graphic construction; and that, for this reason, it will be compulsory to vary the steps, as is well exemplified in the 3rd solution to the Tangencies. FOURTH SOLUTION. (See Plate.) To describe a circle to touch three given circles A, B, C. ANALYSIS. Let O be the centre of the required circle, and let D, E, F be its points of contact with the given circles A, B, C. In OB take OH = OA, and then EH is = DA; and it is evident the circle having B as centre and BH as radius is known. Moreover, if M be the other point in which AH cuts this circle, and K that in which a tangent to it at M cuts AO, then, since BM and AO are parallels, it follows that MK is perpendicular to AO. It is also evident that AO.AK = } AH.AM, and is .*. of known magnitude. Similarly, if in OC we take OI = OA, and that from the other point P in which AI cuts the known circle having C as center and CI as radius, we draw a tangent to cut AO in L, then will this tangent be perpendicular to AO, and will AO.AL = 3 the known magnitude AI.AP. Now if N be the point in which MK cuts AIP, then as AN.AI has to AP.AT the same ratio which AN has to AP or which AK has to AL or which AK.AO has to AL.AO, it follows that AN.AI = twice AK.AO = AM.AH, and .:. that the locus of N is a known circle G having with circle C the point A as centre of similitude. And since PL is tangent to C at P, it is evident KN is tangent to circle G at N; .*., smce MKN is common tangent to the two known circles BM and GN, it isitself known; and AK perpendicular to it is known, as also the point O such that AO.AK = 4 the known magnitude AH.AM. Hence the circle DEF is known. COMPOSITION. Draw any radius BH’ of the given circle B; draw any radius CE’ of the given circle C; in BE’ and CF’ make E’H’ and FI’ each equal to the radius of the given circle A; with B and C as centres and BN’ and Cl’ as radii deseribe circles ; E 2 52 The Three Sections, Tangencies, draw AH’ and AI’ to cut these circles again in M’ and P’; find N’ in AI’ such that AN’.AI’ = AH’.AM’; draw N’G parallel P’C to cut AC in G; with G as centre and GN’ as radius describe a ee ; then, according as — has hke or unlike sign with £8, draw MN a common tangent direct or inverse to the circles BM’ and GN’; draw AK perpendicular to MN and in it find the pomt O such that AO.AK =} AH.AM (this can evidently be done by producing AK until KA’ = AK; and then describing the circle A’M’H’ to cut AK again in O.) The point O is a centre of a required circle, &c. NOTES. Here, too, as in the last solution, it may be remarked that the general solution gives more than one method when applied to many of the particular cases. From this solution also we arrive at that given by Cauchy for two cércles and a point (by supposing the circle A infinitely small), and we arrive at that of Pappus given in Leslie’s Geometrical Analysis for the case of two poimts and a circle (by supposing the circles A and C infinitely small). Moreover, we see what has not been remarked by the authors of these solutions to the particular cases, viz. :—that the perpendicular from the poimt A on -the tangent MN passes through the centre O of the required circle. FIFTH SOLUTION. To describe a circle to touch three given circles A, B, and ANALYSIS. Let D, E, F, be the points of contact of the required circle with A, B, and C. Now (as will appear from some of the porismatic develop- ments), if DN be a perpendicular from D on the radical axis of the circles A and C, and that DM isa perpendicular on the radical axis of the circles A and B, then will DN.AC have to DM.AB one of the four ratios comprehended in that which (AC)?— (rad A + rad C)? has to AB*—(rad A + rad B)?. Hence it is evident DN has to DM a known ratio; and .°. as the radical axes RN and RM are known; it follows that ' THE TANGENCIES Fifth Solution , | i $f. Rie fe E eveett iat vieltncina t fs iP ha. Ca 5 A ghetes! | paene PERE DED Hts RR Ee si He ide : ; ae FERC aU ris wey eee gl. | " SV , ne pity ony y rt y 3 i a 4 3 bh + m ah Leoval tn: tae m i : ae whe / Y Tab. ba rad ek 5 ‘ re wari? and Loci of Apollonius, &c. 53 the straight line RD is known in position; and .’. the point D where it cuts circle A is known. Similarly, by drawing perpendiculars from E on the radical axis RM and on the radical axis to the circles B and C, it can be shown that RE through R is known in position, and .*. E is known. And in like manner we can find the point F on circle C. Or the circle DEF can be easily found from any of the points D, E, F, of contact, since the lines joining these points pass through known centres of similitude, &c. The composition may be easily made. NOTES. The ratio which DN has to DM, is as has been indicated, pn AB {(AC2—Gad A F rad Cp} DM ~~ AC{(AB2—(rad A F rad B”} And, in order to show that this holds good for all values of the radii C and B from zero to infinity inclusive, let ¢ and 4 be the points in which AC and AB cut the circles C and B, and let a be that in which either of these lines cuts circle A. We have AC = Ac + cC, rad A = Aa, rad C= Ce, rad B = Bd; and .:. we can put the ratio under the form DN AB{(Ac’ 42 Ac.cO—Aa” +2 Aa.cC } DM “ac {(Av’ +2.A0,0B—Aa + 2 Aa.bB)} which evidently holds good for all values of the radii B and C. The above method of solution requires one circle (as A) to be finite. When circles C and B are infinitely smalls DN AB Act—Aaz_ AB AC?—Aa2 DM~ AC Ab2-Aa— AC AB—Aa@ When circles C and B are infinitely greats DN A°+ Aa DM Ab+Aa When ¢ is infinitely great and 6 = zero DN at 2.AB (Ac + Aa) DM “AB,—Aai_ When c = zero, and 6 infinitely great pn _ACAw DM 2.AC(Ab+ Aa) From Gergonne’s solution, and the theorem on which the above solution depends, it is evident we have expressions for 54 The Three Sections, Tangencies, the ratios of the perpendiculars from the poles of the axes of similitude on the radical axes, &c., &c. SIXTH SOLUTION. (See Plate.) To describe a circle to touch three given circles A, B and C. ANALYSIS. Let D, E, and F be the points of contact of the required circle with the three given circles A, B, and C. Then DE, DF and EF pass through O, P and Q centres of similitude of the given circles; and OPQ is an axis of similitude. Let G and H be the other points in which ED and FD cut the circle A, and S that in which HG (which is parallel EFQ) cuts POQ. Then OS has to OQ the known ratio which OG has to OH, and which radius A has to radius B, and .. 8 is a known point. Now the ratio OG.DH to PH.DG, being the same with that of OS to PS, is known; and the ratio of PD.PH to to OD.OG is also known; .°. the ratio compounded of these ratios, or that of PD.DH to OD.DG is known: or—which amounts to the same—the ratio of OE.FD to PF.ED bemg the same with that of OQ to PQ is known; and the ratio of PF.PD to OE.OD is known; and .:. the ratio compounded of these ratios, or that of PD.DF to OD.DE is known; and .. as DF: DE :: DH : DG, it follows that the ratio of PD.DH to OD.DG is known. Let I be the other poimt in which a circle through P, H and G would cut OGD. Then PD.DH = ID.DG; and ... ID has to OD the known ratio which PD.DH has to OD.DG, and the point I must be in the circumference of a known circle ILL, having with circle A the pomt O as a centre of similitude. Moreover, if K be the other pomt in which DO cuts this circle, then as the angle LK right to lis = HG right to D, itis = ID right to P; and.:. PI is a tangent to the circle IKL at I. Now PI is known, and .:. also IO and the point D on circle A similar to I on circle ILK, as also the point E on B dis- similar to D on A; and the pomt F on circle C in which PD THE TANGENGIES Sixth Solution gil een ry and Loci of Apollonius, &c. ee cuts it dissimilarly to D on circle A is known. Hence the circle DEF is known. COMPOSITION, Find O a centre of similitude of the circles A and B; find P a centre of similitude of the circles A and C; find Q the centre of similitude of the circles B and C which is in line with O and P; and find the point S in POQ such that PQ : PS =z: rad C : rad A. Take any point D’ on the circumference of circle A, and draw OD’ and PD’ to cut the circles B and C in the pomts FE’ and F’ dissimilar to D’ on circle A; then on OD’E’ find the point I’ such that I'D’ shall have to OD’ the ratio com- pounded of the ratios of OS to PS, and of PD.PH to OD.OG: or—which is the same—find I’ such that I'D’ shall have to OD’ the ratio compounded of the ratios of OQ to PQ and of PF.PD to OH.OD. Draw I’M parallel to AD’ to cut AO in M; from M as centre and with MI’ as radius describe a circle, to which draw PI a tangent; draw IO to cut circle A in D similarly to I on circle M, and to cut circle B in E dissimilarly to D on A; draw PD to cut circle C in F dissimilarly to circle A in D; describe the circle DEF, Then is DEF a required circle. NOTES. This solution holds for all the cases in which the circle A is finite, &c. ' If we were to draw DR tangent to the circle A at D to cut PO in R; then as DR is parallel to PI, and that RP has to RO the known ratio which ID has to OD, it follows that the point R on PO is known; and .°. the tangent RD to circle A is known; and hence, &c.—another method of solution. Or we might solve the problem in a similar manner to that of the third solution from the knowledge that DF.DM has to DE.DL the known ratio of AC? —(rad AF rad C)? to AB’— (rad A = rad B)? where M and Lare the other points in which PD and OD cut the circles C and B. SEVENTH SOLUTION. (See Plate.) To describe a circle to touch three given circles A, B, C, 56 The Three Sections, Tangencies, ANALYSIS. Let O be the centre of the required circle, and D, H, F the points in which it touches the given circles A, B, C. Then ED passes through P a centre of similitude of the circles B and A; and FD passes through Q a centre of simili- tude of the circles C and A. And since the rectangles PD.PE, QD.QF are of known magnitudes, and that they are respectively equal to PO? -— OD? and QO?—OD?, .:. PO?—QO? 1s of known magnitude and sign, and the locus of O is a known straight line OI perpen- dicular to PQ, and such that PI? — QI? = PO? — QO’. _ Now if in OB we have OM = OC, then EM = the radius of circle B; and BM is of known magnitude; and the circle having B as centre and BM as radius is known. Let N be the other point in which OM cuts this last men- tioned circle; and suppose the circle CHL passing through the point C and having with the circle MN the line IO as radical axis. Then L being the other point in which OC cuts the circle CLH, we have OC.OL = OM.ON;; hence as OC = OM, it follows that CL is equal the diameter MN, and .-. of known magnitude. But the circle CHL is known; .«. CL is known in position, and hence the point O in which it cuts IO, and therefore the circle DEF is known. COMPOSITION. Find P a centre of similitude of the given circles A and B; find Q a centre of similitude of the given circles A and C; draw a straight line PD’E’ cutting circles A, B, in dissimilar points D’ and E’; draw astraight line Qd/’ cutting the given circles A, C, in dissimilar points d’ and f'; find the point I in the line PQ such that PI? — QI? = PD’.PH’ — Qd'.Q/’, and through I draw a straight line IR perpendicular to PQ: draw any radius Be’ of the circle B, and from e’ in the proper direction on e’B make e’m’ = the radius of circle C;{ with B as centre and Bm’ as radius describe a circle, and produce e'm' to cut it in n’; through the pomt C describe the circle CHL which with circle Bm’n' has IR as radical axis, and in it inflect the chord CL = to the diameter m‘n', and let O be the point of intersection of CL with IR: then will O bea centre of a required circle. — oe “ poe forty ‘ Fs ale by , A De: > i ae: hate. ene: I Fig ved ore Tit SR , . ie : ; ae q nasties. t.. ae wil? Nha ‘wy tig HP, “2 ] =~ 2 eine couarinnih ee Bs of ee —~ 2 i, fh icaivdl Obixio iia Ory Pe i i): 7% iio ach basal mils Gi at teres? > | oni gh bedtiat ti” Bs reel eh arma led suse re 5 oe 74 Pre fe it . = THE TANGENCIES Eighth Solution. and Loci of Apollonius, §c. 57 NOTES. Since there are two points P and two points Q, there are four lines IR, and as there are four corresponding circles CL and two chords CL in each, it is obvious there are eight answerable circles O real or unreal, in pairs, according as the circles LCM are greater and less than the corresponding circles m’n’. It is evident that when the circles A and C are infinitely small, and B finite, then may the point Q have any position whatever in the straight line through A and B (because the infinitely small circles may have any ratio whatever just according as we suppose two circles to have any finite constant ratio during their diminution to the infinitely small state.) This solution does not readily apply to the case in which two of the given circles are supposed infinitely great, or replaced by straight lines ; but the following is an analysis of a solution which will embrace all the cases in which we sup- pose the circle A of finite magnitude. Since the rectangles PD.PE and QD.QF are known in signs and magnitudes, it follows (from one of a class of po- risms to be included in subsequent developments) that we know the two points o’, o', of real or imaginary intersection of all circles having their centres in PQ and respective radii equal the tangents from them to circle DEF. And since the circle A and straight lme PQ are known, we know the two points a’, a’, of real or imaginary intersection of all circles having their centres in PQ and respective radii equal to the tangents from them to circle A. And it is evident we know the circle through the four points o’o’a’a’, and that its centre R is in PQ; moreover it is evident the cir- cumference of this circle R passes through D; and .-. D on circle A is known, and hence AD and the point O in which it cuts the straight line through o’o’; and .:. the circle DEF is known EIGHTH SOLUTION. (See Plate.) To describe a circle to touch three given circles A, B, C. ANALYSIS. Let O be the required centre, and let D, E, and F be the points of contact with the given circles A, B, and C. 58 The Three Sections, Tangencies, Since the rectangle under the half sum and half difference of the sides of a triangle is equal to the rectangle under the half sum and half difference of the segments of the base made by a perpendicular from the vertex; .*. it is evident, that if OP be perpendicular to AB, and M be middle poimt of AB, and that in OB we have Oe = OA, and 4 the middle of eB, then will 06.B6 = AM.MP;; and hence Od has to PM the same known ratio which AM has to OB. Now if in AB we find the pomt G such that eb is to GM in the known ratio of Ob to PM, then will Oe or its equal AO have to PG the same known ratio. For like reasons it is evident that if OQ be perpendicular to AC, and N the middle poimt of AC, and that we assume Of in OC and = to OA, and that ¢ is middle of FSC, and that we find H in AC such that fe shall have to HN the known ratio of AN to Cc, then will AO have to QH this same known ratio. Now the points G and H are known, and the ratio of PG to QH is known (because AO has known ratios to PG and QH): hence the pomt I in which the circle AHG cuts the circle QAP is known; and since the angles P and Q are right, it follows that the straight line IO perpendicular to AI is known, as also the pomt Rin which it again cuts the circle AGH Again, GR bemg perpendicular to AG, it is parallel to PO; and PG has to OR a known ratio; therefore AO has to OR a known ratio. If 8 be the pomt in which AO again cuts the circle AGH, it follows, from similar triangles, that AI has to RS the known ratio which AO has to OR, and .*. the chord RS is of known magnitude, and .:. also it is known in position; and AS is known, and also the point O where it cuts IR, and .-. the required circle is known. COMPOSITION. Through the centres B and C draw BH’ and CF’ any two radii of the circles B and C; from E’ in either direction on EB make E’e’ = radius of circle A; frem F’ in either direc- tion on F’C make F’f’ = radius of circle A; bisect e’B in @’, Cf inc’, AB in M, and AC in N; find Gin AB such that e'b' : GM :: AM: ed’; and in AC find the pomt H such that fic’ : HN :: AN : fc’; assume any straight line x and find lines y and z such thatw : y :: AM: eb’, anda: z Sf bi vt 7 ees v's mere pew ag Pee dase) tye nis syed ly ih He [id os Mo fin wan ue Se sai Aare Dy pr see oY Tene ee eich oP eS ep ing it “y¢4 PE Mii Fats oe bai ty: at Are 3 a by Sank! onde pel ap Menno: | it: apy: ait, ki inte x oaky Mie 32 . ‘ 7 : 7 7 +4 ra ; “4, ig ae at 2 Pio Avy a cs bie be hirer beige er yea Pe a Bee Sa Ree 1 ‘ ; 7 " 3 uy - ¢ ‘ : a oy 4 a , : ; an,” P r ‘ y pathy ship oo mts laa Arey agreed eet ret Ae Peak a * ay : Fe ORS AG Ee se re WE eae REE Vy . sega rereyis hs Mrua 3 Hen of 2 : . Las , yt : * i i Pie a 4>¥% i. 7 oe ; : } 1a Hak 9 Ler aoa oh , ae fe, ec pa? : ; wlittes sh ‘ ‘ trl we, © z Wy \“e 2 bs . os * , vr, Se A Om vy ; 5 ' 7 , r ~~ * i ) “ bor 2) ae rn yrs eit - 4 i ¥ ry SED RAS TEA " , i CREE in A Peeks ol bai ‘ e's a r Pits: Beran » bah THE TANGENCIES Ninth Solution and Loci of Apollonius, &c. 59 :: AN :: fe’; on AB and AC make Gp and Hg equals to and z, and describe the circles AGH, Apg; through the point of intersection I of these two circles draw IR perpendicular to AI to cut the circle AGH again in R; find the point 2 in IR so that AI : Ti :: AM : e’d’, and draw i’ parallel to AG to cut AI in #’; with R as centre and radius equal 2’ describe a circle, and from either point S in which it cuts the circle AGH draw AS to cut IR in O. Then will O be the centre of a required circle. NOTES. This method of solution holds good in all cases in which circle A is not supposed infinite, or—which amounts to the same thing—it holds good in all cases but those in which we suppose the three given circles replaced by straight lines. If we suppose circle C infinite, then it is evident AC is parallel to OC and perpendicular to tangible portion of the infinite circumference; moreover since the ratio of AN to fe is then one of equality, so will that of AO to QH be one of equality, and ... QH will be = Of, and the pomt H im the perpendicular from A on the known portion of the infinite circumference is known as it is at a distance = radius of circle A therefrom. It may also be observed that the line [OR is identical with the line IOR of the seventh solution. NINTH SOLUTION. (See Plate.) To describe a circle to touch three given circles A, B, C. ANALYSIS. Let O be the centre of the required circle, and let D, E, F be its poimts of contact with the given circles A, B, C. If in OA we suppose OM taken equal OB and so that OB and OM have like directions in respect to the directions OK and OD, then DM = KB, and it is evident AM is of known magnitude. And since OB is equal OM and that 2.0A.0OM = OA? + OM? — AM?. *, 2.0A.0M = OA?* + OB? — AM? both in sign and magnitude. But if G be the point in which the circle having 60 The Three Sections, Tangencies, BA as diameter cuts AO, we have 2.0A.GO = — OA? — OB? + AB?. And from these two we get, by adding equal quantities, the relation 20A.GM = AB? — AM’. Similarly if in OA we suppose ON equal to OC and the like direction in respect to direction OD which the direction OC has to OF, then will DN = FC, and will AN be of known magnitude. And it is evident that if H be the pomt in which the circle having CA as diameter cuts AO, we have in like manner the relation 20A.HN = CA? —AN?. But the ratio of the known quantities AB? — AM? and CA? —AN? is known: .°. it follows that the ratio of GM to HN which is the same with it is known. Now if in MN we suppose NL so taken that MA has to NL the known ratio which GM has to HN; then GA has to HL the same ratio. Hence if in AO we suppose KA = HL and in like direction to it, then GA has to KA a known ratio, and .-. the point K must be in the circumference of a known circle AKP passing through A and having its diameter AQ in AB. Again since KA = Hh, we have KH = the known magnitude AL. And the angles CHK, QKH, beimg right, it follows that QT the perpendicular from Q on CH is equal to KH, and that CT is tangent to the circle having Q as centre and QT as radius; but this circle is known; there- fore the tangent CT is known, as also the other pomt H in which it cuts the circle on AC as diameter; and therefore AHO is known in position. And, since AHO is known in position, the point N is known; and ... as OC = ON, the point O is known, and hence the circle DEF. COMPOSITION, Through the centre A draw any radius AD’; in D’A take D'M’ = radius B, and D’N’ = radius C; im M’'N’A find the point L’ such that M’A shall have to N’L_ the ratio which AB? —AM” has to CA? — AN” (taking signs into account) ; in AB take AQ so that AB: AQ:: M’A: N’L (taking note of signs) ; on AC and AQ as diameters describe circles ; with Q as centre and a radius equal to AL’ describe a circle, and draw CT a tangent to it from C ; through the other point H in which CT cuts the circle on AC as diameter, draw the straight lime AH to cut the circle on AQ as diameter in K ; make HL = KA (and in the same direction) ; make in the Rt at ime exer aE. Nie ithe “f be ieee fe TS Wt Bea ae ve THE TANGENCIES APOLLONIUS ORIGINAL SOLUTION. and Loci of Apollonius, §c. 61 same line AN = AN’ so that 44 and AX’ have like signs; find O in Ali so that OC = ON; then from the point O as centre and the point D, when NO cuts circle A (so that ND = N’D’) as distance, describe a circle : this circle will be as required. NOTES. This method of solution is not intelligibly applicable to those states of the data in which any of the given circles is supposed infinitely great, or replaced by a straight line; but for the other cases it is thoroughly complete and deserves attention. Indeed, I may remark that there are many very good solu- tions applying only to the cases in which none of the given circles A, B, C, is infinite, or when two are infinite. Yet it must not be lost mind of, by those who would succeed, that the solutions to general questions are often arrived at from considering them under some particular states of the data, and divining what modifications are necessary so as to make the solutions which may be arrived at applicable to the more general cases. It may also be remarked that a theorem evolved in the above solution is directly applicable in a solution to the prin- cipal case of the ‘ Jnclinations’ of Apollonius. APOLLONIUS’ ORIGINAL SOLUTION. . (See Plate.) To describe a circle to touch three given circles A, B, and C. ANALYSIS. Let D, E, and F be the points of contact of the required circle with the given circles A, B, and C. Then, DE passes through O a centre of similitude of the circles A and B; DF passes through P a centre of similitude of the circles A and C; and EF passes through Q a centre of similitude of the circles B and C; and the points O, P, and Q are in a straight line. Let G and H be the other points in which DE and DF cut the circle A. Through H draw a parallel to OPQ; and through the other point I in which it cuts the circle A, draw DI to cut OPQ in M. 62 The Three Sections, Tangencies, Since HI and HG are parallel to FEQ and QPO, the angle QO right to E = angle HI right to G and is ... = angle DI right to G or DM right to O or H; and .-. a circle can pass through MDEQ. But O being a centre of similitude to the circles A and B, the magnitude of OD.OE is known; and .:. since OM.OQ = OD.OE, the point M is known. Let N be the poimt in which a tangent to circle A at H cuts POQ. The angle NH right to Pis = HN right tol and... = DH or DP right tol or M. Hence acirele can pass through DH, M, and N. Andsince PM.PN is = the known magni- tude PH.PD, it follows that the pomt N is known. Now N bemg a known point, the tangent NH to circle A is known; and the pomt F im which PH cuts the circle C similarly to H on circle A is known; as also the other poimt D in which it cuts circle A. And the pomt E on circle B, in in which OD cuts it dissimilarly to D on circle A, is known: and hence the circle DEF is known. Or, we might determine D by drawing a tangent at D to cut OPQ, &c., because circle MDP touches A at D. COMPOSITION. Find O a centre of similitude of the circles A and B; find P a centre of similitude of the circles A and C; and find @ the centre of similitude of the circles B and C which is in line with O and P. Through O draw a straight line to cut the circles A and B in dissimilar pots, D’ and EH’; describe the circle QH'D’; through P draw a line to cut the circle A in pomts H’, D’; through H’, D’, and the other pomt M in which the circle QE'D’ cuts the line OPQ, describe a circle; through the other point N, in which the circle D'H’M cuts the lime QPO, draw NH atangent to the circle A; draw PH to cut the circle A again in D, and to cut the circle C in F dissimilar to D on A; draw OD to cut circle B in E dissimilar to D oncirele A; describe the circle DEF. Then is DEF a required circle. NOTES. This method of solution is intelligibly applicable im a direct manner only when the circle A is finite, and C neither an and Loci of Apollonius, &c. 63 infinitely great circle, nor infinitely great in respect to the circle B. The reason of such restriction arises from the peculiar nature of infinitesimal geometry causing the indica- ted operations to be graphically impracticable though men- tally possible. However as regards the five principal cases of the problem; viz., when the circles A, B, C are finite—when A and B are finite and C infinitely small—when A is finite and B and C infinitely small—when A and C are finite and B infinitely great—and when A is finite, C infinitely small and B infin- itely great, this solution is remarkably elegant, and depends on very simple well known elementary truths. That it is in substance the same as the one given by Apollo- nius, may be easily gathered from Pappus’ commentaries on the writings of the celebrated Greek geometers. He observed that the Apollonian solution to the Tangen- cies was of such a uature as to indicate a method of inseribing a triangle in a given circle, whose sides would pass through three given points in a straight lime. And then, evidently, in order to prepare for a construction to the general problem of inscribing, in a given circle, a polygon, whose sides should pass through given points, he gives the indicated method of solution to the particular case just mentioned, both when the three fixed points are at finite distances from each other, and when one of them is at an infinite distance. Now, in the solution just given in the text, nothing would be more apt to suggest itself than the fact that GH, a parallel to FEQ, cuts OPQ in a point §, such that OS has to OQ the known ratio which OG has to OF, or which rad. A has to rad. B; and that we could .:. solve the problem :—“ Being given three points O, P, 8, ina straight line; to inscribe a triangle DGH m a given circle so that its sides will pass through these points.” | And the method which the present solution indicates is exactly the same as is given in Pappus’ Mathematical Collec- tions, as well when the three points are at finite distances as when one of them is at an infinite distance. These coincidences in peculiarities are, I consider, sufficient to justify me in believing that I have reproduced the solution of the celebrated Greek geometer. And I feel the better pleased at this as it clears up along disputed point concerning the claims of the rival’ ‘restorations’ given by Vieta and Simson to the case of the problem in which one of the cireles as C is infinitely small. » 64 The Three Sections, Tangencies, It will be seen that Vieta’s solution, in the most improved form, is the same as that of Apollonius. And here, before closmg my notes on this celebrated problem, I may observe that Dr. Robert Simson, like many others, certainly misunderstood the object of propositions 116, 117 and 118 of Book VII. of Pappus’ Mathematical Collections ; and that through this he was led to imagine he restored or reproduced the proposition to which they were intended as subsidiary. However, as Dr. Simson’s remarks are interesting in a historical point, I will give them as translated from the Appendix to his Opera Reliqua, by Professor Davies. They are as follows :— ‘‘In the Seventh Book of the Mathematical Collections of Pappus Alexandrinus (every admirer of the ancient geome- trical analysis ought to rejoice that this work has been preserved to our times), among the lemmas which that most eminent writer has handed down, there exists a problem for one of the tangencies of Apollonius, namely, m Prob. 117, B. VII; in which it is required, when a circle being given by position and three points in a straight line, to inflect from two of the points two lines meeting in the circumference, so as to make the two points in which they intersect the circle and the third given poimt in the same straight line. It is not difficult to investigate the rest of the lemmas which are sub- sidiary to the problems on the tangencies ; and some of these Vieta has used in his Apollonius Gallus ; but to what problem the aforesaid lemma could be subsidiary, neither Vieta nor any other geometer has attempted to conjecture. “‘ Often, indeed, have I resolved the subject in my mind, but I have never succeeded in arriving at any satisfactory conclusion, except that the lemma, by no uncertain marks, appeared to be necessary for the following problem :—Two circles and a pot being given by position, it is required to describe a third circle which shall touch the given circles and pass through the given pomt. In what manner, however, the lemma might be subsidiary to this problem I did by no means perceive. I have directed my attention to the solu- tions of Vieta and others, hoping that by chance I might hit upon the analysis requiring this lemma, but in vain; until this day, after various trials, I discovered the true analysis of Apollonius, to which, indeed, both this Prob. 117 of Pappus, as well as Props. 116 and 118 are manifestly subsidiary.— February 9, 1734.” and Loci of Apollonius, &e. 65 How such an able geometer could look so long in vain for a solution to the tangencies which might implicate the 117th proposition of Pappus’ 7th Book, I am at a loss to understand ; though it evidently accounts for his implied opinion that the general problem of the three circles was originally referred to the particular case of two circles and a point. Indeed, J may mention that the solution which I give as that of Apollonius was the first one which suggested itself to me for the general question of the three circles; though not exactly in the form in which I now present it: for after arriving at the point in the analysis in which M is shown to be found, I proceeded as follows :— Since HI is parallel to PM, a circle through M, P and D touches the circle A im D; and .-., sce P and M are known, this touching circle MDP is known; and hence ODE, PDF, and circle DEF are known. I may further note, that we may give another method of solution implicating Pappus’ lemma, by supposing K the point in which DK parallel to OPQ cuts circle DEF, and V that in which FK cuts OPQ. For as the angle FE right to K = DE right to K, it is OE right to V and ... QV.QO being = QE.QF, the point V is known. And if T be the point in which the tangent at D cuts OPQ; then since the angle FK right to D = DK right to T, it is— TP right to D, and .:., PT.PV being = PD.PF, the point T is known. Hence the tangent TD to circle A is known, and .-. ODE, PDF and circle DEF are known. It is evident this method holds graphically good only when A is finite and C neither infinitely small, nor infinitely small in respect to the circle B. It applies to the case in which A is finite and B and C infinitely greats; but does not to that in which A is finite and B and C infinitely smalls. Similar solutions to the two just indicated are obvious from the ‘ Involution’?’ Theory as unfolded in the Géométrie Supérieure :— 1. Let D, E, F be the poimts of contact with the given circles, and OPQ the known centres of similitude, in a straight line, through which DE, DF and EF pass. If G and H be the other points in which DE and DF cut the circle A, then GH is parallel to EF, and OG : OE :: rad A : rad B; and hence § the point in which GH cuts OQ is known. Now, if we suppose T the point in which atangent to the circle A at D cuts OQ; then, since we may regard GHDD as F 66 The Three Sections, Tangencies, an inscribed quadrilateral mm circle A having the side DD infinitely small, it follows that the straight line OPST is cut in “involution” by the circle A and the pairs of opposite sides of the quadrilateral; but the circle A is known, as also the poimts O, P and 8; therefore the pomt T can be found as follows (see Chasles’ Géométrie Supérieure, page 150) :— Assume any point U on the circumference A, and describe the circle OPU; through U, 8, and the other pomt V in which the circles A and OPU intersect, describe a circle UVS: then will this circle UVS give the poit T in its other intersection with OPS. Hence we know the tangent TD to circle A, &c. 2. We can find the tangent NH im a similar manner. Thus :—Through W any point in the circumference A describe the circle PWS; through O, W, and the pomt X in which this circle cuts the circle A describe a circle: then will this circle give N in its other intersection with the line SPO. Hence the tangent NH, &c. . 3. Or, since the required circle DEF evidently cuts OPQ in known points, real or imaginary, it is obvious that if through any point D’ we draw two straight lines OD’ and PD’, and on them take D’E’ and D’F’ such that OD’.OE’ = OD.OE, and PD’.PF’ = PD.PF, then will the circles D’E’F’ and DEF have OPQ as radical axis; and we can find the tangent DT as follows :—Describe the circle OPD’, and through Q, D’, and the other point K of intersection of the circles OPD’ and D’E’¥’ describe a circle: then will the other pomt im which the circle D'KQ cuts OPQ be the required pomt T from which to draw the tangent TD to the circle A. Montucla gives a very curious history of this problem. He says:—“ Vieta, in a dispute with Adrian Romanus, proposed this problem. The solution given by Romanus, though obvious, was very indifferent, viz., by determining the centre of the required circle by the point of intersection of two hyperbolas. Vieta solved it very elegantly in his Apollonius Gallus, printed at Paris in 1600: his solution is the same as that given in Newton’s Universal Arithmetic. Another solu- tion may be seen in Lemma 16, Book I., of the Principia (this question being there necessary for some determinations in Physical Astronomy), where Newton, by a remarkable dexterity, reduced the two higher loci of Romanus to the intersection of two straight lines. Moreover, Descartes attempted to solve this problem by algebraic analysis, but without success; for, of the two solutions which he derived and Lovi of Apollonius, &c. 67 from thence, he himself acknowledges that one furnished him with so complicated an expression, that he would not under- take to construct it im a month; while the other, though somewhat less complicated, was not so very simple as to encourage him to set about the construction of it. Lastly, the Princess Elizabeth of Bohemia, who it is well known honored Descartes with her correspondence, deigned to com- municate a solution to this philosopher; but, as it is deduced from the algebraic calculus, it labours under the same incon- veniences as that of Descartes.” Euler, Fuss, T. Simpson, and other eminent analysts have given algebraic solutions, though not at all commensurate with the requirements of the problem. T. Simpson has also given a geometrical solution in the appendix to his Elements of Geometry, which, in reality, does not differ in principle from Newton’s in the Principia. These solutions, like those of Euler, are very imperfect, though complete ones of a similar nature, and much more simple, can be easily formed. The late John Mulcahy, L.L.D., professor, Queen’s College, Galway, after giving Gaulthier’s solution as improved by Ger- gonne, in article 68 of his Principles of Modern Geometry, again returns to the subject in article 95, and deduces Gaul- thier’s original method depending on the circle which cuts the three given ones orthogonally: this of course labours under the disadvantage of being applicable when the radical centre is within the three given circles. Those who are acquainted with the Principles of Modern Geometry, or the writings of the late Professor Davies, of the Royal Military Academy, will at once see that all my methods are applicable when, instead of three circles in a plane, there is given three circles on the surface of a sphere. 'The only differ- ence being that straight lines, whether im data or solution, will be represented by great circles of the sphere. My solutions have also analogous ones answering to the following celebrated problem, which was proposed by Des- cartes to Fermat:—‘“‘ Suppose four things, A, B, C, D, to be given in position, consisting of points, planes, and spheres, which may be taken of any one of these kinds exclusively, or of any two of the kinds, or of all of the three kinds ; it is required to describe a sphere which shall pass through each of the given points, and touch the given planes or spheres.” All we have to do is (as in considermg the Apollonian problem) to form the solution for the general case im which F2 68 The Three Sections, Tangencies, the data is four spheres, and then make the modifications necessary when we suppose one or any number of the spheres to become infinitely small or infinitely great, or—in other words—when we suppose one, two, three, or all of the spheres to be replaced by points or planes. The analysis similar to my first solution is evidently as follows :— ANALYSIS. Let a, b,c, d be the respective poimts of contact of the required sphere with the given spheres A, B, C, D. Then the straight lines ad, ac, ad, pass respectively through O, P, Q, known vertices of similitude to the pairs of spheres AB, AC, AD. Now, if a’ be any point in the surface A, and that 0’, c’, d, are the respective points in the surfaces of B, C, D, made by the straight lines @‘O, a’P, a'Q which are dissimilar to that of a’ on the surface of A, it is evident Oa'.0b' = Oa.08, Pa’.Pc’ = Pa.Pe, and Qa’.Qd' = Qa.Qd, and therefore that the spheres a’‘b‘c'd’, abcd, have the plane OPQ as radical lane. : This being borne in mind, it is evident that the tangent plane to any of the given spheres at the poimt of contact will cut the plane of section of this sphere and the sphere a‘b'c'd' in a straight line situated in the plane OPQ. But as we may assume the point a’ anywhere in the surface of A, we know the resulting points 6’c’d’, and the sphere a‘b'c'd', and its planes of intersection with the spheres A, B, C, D, and also the intersections of these planes with plane OPQ, and the tangent planes from these lines to the given spheres, and .*. the sought sphere of contact. And it may be remarked that to each plane of similitude OPQ there are two answerable spheres abcd whose centres (as also the centre of the corresponding sphere a’‘b'c'd’) are on the perpendicular from the radical centre of the four given spheres to the plane OPQ; and, moreover, that as there are eight planes of similitude OPQ there are sixteen answer- able touching spheres (real or unreal in pairs). We ma further remark that this solution furnishes a proof that the twelve vertices of similitude of the four given spheres lie in sixes in the eight planes OPQ, and are the vertical points of a complete octahedron. It is also easy to see that the chords aa of the sphere A and Loci of Apollonius, &c. 69 pass respectively through the poles of the corresponding planes OPQ in respect to sphere A, and all through R the radical centre of the four given spheres ; and therefore, it follows, that the eight lines in which the planes of intersection of the eight spheres a’b‘c'd' with the sphere A cut the corresponding eight planes OPQ of similitude, are situated in the polar plane of the radical centre R in respect to the sphere A, &c. Or we might determine the point @ (and hence 48, ¢, d) from the following considerations :—Since Oa.Ob, and Pa.Pe are of known magnitudes and that aO.ab has to aP.ac a known ratio, therefore the point a must be on the surface of a known sphere haying its centre in the straight line through O and P. Similarly, since Pa. Pe and Qa.Qd are of known magnitudes and that aP.ac has to a@Q.ad a known ratio, therefore the point @ must be on the surface of a known sphere having its centre in the straight line through P and Q. Hence, as the point @ is on the sphere A, it follows that it must be a point of intersection of the circular traces made on the sphere A by the two known spheres having their respective centres in the straight lines PO and PQ. The other solutions to Descartes’ problem of the spheres, which are analogous to those I have given to Apollonius’ problem of the circles, may be easily made :—the tangent to two circles in the plane being represented by a plane touching three spheres, &c. And the actual operations are very simple when performed according to Monge’s practical processes of the geometry of figured space, known by the name of “ De- scriptive Geometry.” APOLLONIAN LOCI PROBLEM. Given two points A, B, and the magnitudes of four lines a, b, ec, d, to find the locus of a point P, such that AP? + ab ; 70 The Three Sections, Tangencies, BP? + ed :: m : n,—/the ratio — being known in sign and magnitude, and the rectangles a.b, c.d being also of known magnitudes and signs). ANALYSIS. Suppose the points C and D on AP and BP such that CA.AP = a.b, and DB.BP = c.d; then we have AP.CP : BRE DPR esis ‘mint But if Q be the poimt in BP such that the angle QA right to P=CP right to D, we have ae ue == AP: CP-sand therefore it follows that QP : BP : nN. Hence if we Ben Ge parallel to ‘AQ to cut AB in O, we have AO : BO: : n, and therefore the point O is known. Moreover, the oie PO right to Q or D bemg equal QA right to P or D, it is equal to CP right to D, and therefore OP is a tangent to the circle CDP. Now it is evident that if P’ be any point whatever, and that C’ and D’ are taken on AP’ and BP’ so that C‘A.AP’ = CA.AP, and D’B.BP’ = DB.BP, then will the circles PCD and P’C’D’ have AB as radical axis, and therefore (as O is a point on this radical axis) the tangent OP equal to the tangent from O to the circle P’C’D’; but the circle P’C’D’ is known; therefore OP is of known magnitude, and_ hence the locus of P is a known circle having O for centre. COMPOSITION, Find O in AB such that AO : BO :: m : n; assume any point P’, and in the straight lines AP’ and BP’ find the points C’ and D' such that C’A.AP’ = 0:6, and DBP aed; describe the circle P’C’D’, and draw a tangent to it from O ; with O as centre and this tangent as radius describe a circle ; this circle will be the required locus. For let P be any poimt in its circumference, and C and D the points in AP and BP such that CA.AP = C’A.AP’, and DB.BP = D’‘B.BP’; and let Q be the pomt in which AQ parallel to PO cuts PB. The straight lime ABO is evidently the radical axis of the circles PCD and P’C'D’, and therefore OP is a tangent to the circle PCD. Hence angle CP right to D being equal angle PO right to D, it is equal QA right to D or P, and therefore and Loci of Apollonius, &c. 71 me.CP: = OP DP... But QP.DP:BP/DP’:: QP: BP ?: moO > BOL mm: a; hence AP.CP. = BP.DP : *m- 2 x, and since AP.CP = AP? + a.b, and BP.DP = BP? + c.d, therefore AP? + 4.6 : BP? + ed :: m: n. DISCUSSION It is evident the locus is real only when the centre and radius are real; and it is also evident the centre is real for all real values of “, and that the radius is real only when the point O does not lie within the circumference P’C'D’. It is evident the circle P’C’D’ will cut AB in real points when a. and c.d are both positive, or when one of them is positive and the other negative; but when a.d and c.d are both negative, as the pomts P’, C’, D’ are then on the same side of AB, it is obvious the circle P’C’D’ will or will not cut AB in real points according to the conditions of the data. And if f and f represent the points in which the circle P'D'C’ cuts AB, it is evident that fA.Af = ab, and that [B.Bf = cd, the equalities bemg im signs as well as magni- tudes. When a.b and c.d are both negative. Putting s and ¢ to represent the magnitudes of mean pro- portionals between a and 8, and e¢ and d, it is evident that when the distance between the given points (disregarding directions) is less than the sum of s and ¢, or greater than their difference, the circle P’C’D’ will not cut AB im real points ; and when the distance between A and B is equal to either the sum or difference of s and ¢, the circle P’C’D’ will cut AB in real coincident points ; and when this distance is greater than the sum of s and ¢ or less than their difference, the circle P’C’D’ will cut AB in real distinct points f, f. When the circle P’C’D’ does not cut AB in real points, it is evident the tangent OP’ is real, and therefore that the locus is a real circle for all real values of — positive or negative. When the circle P’C’D’ touches AB, the tangent OP’ is always real, and in this case too the locus is always real : moreover, it is evident that when m:n:: Af: Bf, the 72 The Three Sections, Tangencies, centre O coincides with the double point f and the locus is an infinitely small circle. When the circle P'C’D’ cuts AB in real distinct point f and f; it is to be remarked that these points are both between A and B, or both on the same side of A and B, according as the distance between A and B is greater than the sum or less than the difference of s and ¢; moreover, when the distance between A and B is less than the difference of s and ¢, the two points f and f will lie in the production of AB through A when sis less than ¢, and in the production through B when ¢ is less than s. And it is obvious that the locus is real for any value of ~ to which the corresponding centre O does not lie between f and /. Hence, when AB is greater than s + ¢, the locus is real for all positive values of =; and, for negative values of =, the locus is a real finite circle, a real finitely small circle, or an imaginary circle, according as = is not comprehended between the limiting values Be and is or equals one of them, or is comprehended between them. Similarly, when AB is less than the difference of s and ¢, it is evident the locus is real for all negative values of ; and for positive values of = the locus is real and finite, real and infinitely small, or imaginary, according as — is not compre- hended between, or equal to one of, or is comprehended between the limiting values af and oe In the case in which c.d = zero, the point D’ coincides with B and the circle C’P’D’ cuts AB in B and in another point f which lies between A and B, or coincides with B, or hes in the production of AB through B according as s is less than, or equal to, or greater than AB, and therefore the limits are a known negative value and infinity negative when s is less than AB, and a known positive value and positive infinity when s is greater than AB. .. when sis less than AB the locus is real for all positive values of =, and also for the nega- tive values of ~ which are not comprehended between the negative limit and negative infinity ; and when s is greater than AB the locus is real for all negative values of ~, and for the positive values of — not comprehended between the posi- tive limit and positive infinity ; and when s = AB the locus and Loci of Apollonius, &c. 73 is real for all real values of ~. And in any of these three cases it is evident the locus is an infinitely small circle when is equal either of the limiting ratios of the case. In case a.b = zero, and c.d = zero; then the points C’ and D’ coincide with A and B, and the points/ and f also coincide with A and B. And it is evident the limits are equal zero and infinity, and that according as any value ~ is positive or negative, so will the locus be real or imaginary; when is equal either of the limits, the locus is an infinitely small circle coincident with one of the given points. If the pomts A and B coincide, and that a. = c.d, and that “=+1; then since the point O may have any position ‘whatever in the line through A and B, and that the circle P’C'D’ is real, it follows that there are innumerable circles such that any pot P m the circumference of any of them fulfils the conditions of the locus. The problem in this state of the data is said to be “ porismatic.”’ When we suppose ~ = + 1, the centre O of the locus is n infinitely distant (when the points A, B are distinct) and the tangent to the circle P’C’D' is parallel to AB ; and it is evident the perpendicular from the point of contact on AB is entirely in the infinitely great circle constituting the locus, and passes through o’ the center of the circle P’C’D’. In this case we have AP? — BP?=c.d — a., and the problem can be enunciated as follows :—Given two points A and B to find the locus of a point P, such that the square of its distance from A, one of the given points, minus the square of its distance from the other, shall be of a given magnitude c.d—a.b of known sign. And if we suppose I a pomt in the line AB such that BLIB = c.d, and that I remains fixed when B is infinitely distant ; then it is obvious that P’D’ and PD are parallel to AI and bisected by the perpendicular to AB through I; and it is evident that when ~ is of a finite magnitude, this perpen- dicular through I is entirely im the infinite locus. Now it is further evident that if we suppose n always = BI, and that B is at infinity in the direction AI, then smce AP.CP : BP.DP :: m: BI, and that for points P at finite distances, BP = BI, therefore AP.CP = DP.m. And it is obvious that as DP is double the perpendicular from P on the perpendicular through I to AB, the problem becomes tantamount to what is given in the enunciation :— 74 The Three Sections, Tangencies, “ To find the locus of a point P such that the sum of the square of its distance from a given point A and a given negative mag- nitude a.b. shall be equal to the rectangle under its distance PK from a given straight line IK, and a line 2.m of a given magnitude whose sign is known in respect to the direction of the perpendicular to the given line 1K.” And since AO : BO :: AP.CP : BP.DP and that, in this case, BO = BP, therefore AO.DP = AP.CP = DP.m, and hence AO = m. Therefore the solution for the problem just enunciated may be worded as follows :— In AI perpendicular to IK find O such that AO = 4 (twice m) =m; assume any pot P’, andin AP’ find C’ such that AP’.C’A = a.b.; draw P’K’ perpendicular to IK, and extend it on the other side of IK until K’D’ = P’K’; describe the circle C’P’D'; draw a tangent from O to this circle; from O as centre with this tangent as radius describe a circle, and it will be the required locus. It is to be observed that the locus is real and finite, real and infinitely small, or imaginary according as O is outside, on, or inside the circle C’P’D’. It is evident that when—a.6 is greater than AI? the circle C’P’D’ aoes not cut AI in real poimts f and f, and therefore that the locus is real for all real values of m. When—a.b = AI the circle C’P’D’ touches AI in I, and the points f coincide in I; and the locus is always real for real values of m, and infinitely small when m = AI. When a.é is less than AI? the circle C’P'D’ cuts AI in real points f and f equidistant from I, and on the same side of A as 1; and it is evident the locus is real for all values of m having a different sign to the direction Al; and, moreover, for a value of m having like sign with direction AI, it is evident that according as it is not comprehended between the limits Af and Af, or is equal one of them, or is comprehended between them, so will the locus be real, real and infinitely small, or imaginary. Further, if we suppose a.b = zero, the limits Af and Af are evidently = zero and twice AI, and include the values of m for which the locus is imaginary. When a.b is positive and c.d negative. Here it is evident the poimt C’ is on a different side of AB with the points D’, P’, and that the circle P’C’D’ cuts AB, in one point between A and B, and in another in the and Loci of Apollonius, &c. 75 production of AB through A. Hence, we see that the limits = and #4 ~ are the one positive and the other negative. And it is evident that for all positive values of ~ the locus is real, or real and infinitely small, or imaginary, , according as is greater than the positive limit Bo or equal to it, or less than it. And for negative values of = it is evident the locus is real, real and infinitely small, or imaginary, SCS as“ is numerically greater than the negative limit + ‘By - or equal to it, or less than it. If c.d = zero ; one of the points f coincides with B, and one of the limits = infinity; and the other point / is in the pro- duction of BA through A. In “eS case the locus is imaginary for all Pee negative values of “, and real only for aoe values of ~ which are not less than the other limit 4 = BF; and when ~ is pail either limit, the locus is real and infinitely small. If a. and c.d be each equal zero ; then one point f coin- cides with A, and the other with B, and the limits are + zero and — infinity, and the locus is real only for finite values of ~ which are positive. And, by similar considerations as were made use of when a.b and c.d were regarded as negative, we are led to know that a particular case of this problem may be enunciated as follows :—‘ Given a point A and a straight line 1K in position; to find the locus of a point P, such that AP? + a given positive magnitude a.b shall be equal to the rectangle under a given length 2.m, and the perpendicular distance PK of the point P to the given line 1K. In which the sign of 2.m in respect to the direction of PK is given.” The solution may be worded exactly as in the similar case when a.6 and c.d were regarded as negative. But it may be remarked that here the points f, f, are always real when the-data is real, and that they are equidistant from I, and on opposite sides of A, &c., &c., When a.b and c.d are both positive. It is evident the points C’, D’, and the point P’ are on opposite sides of AB, and that the circle P’C’D’ always cuts AB in real points f and J, one of which is in each production of 76 The Three Sections, Tangencies, AB. And it is obvious that for all negative values of= (as the corresponding points O lie between A and B) the locus is ima- ginary ; and it is further evident that for positive values of = between zero and the lesser limit, and between infinity and the greater limit, the locus is also imaginary ; and that for all other positive values of = the locus is real. And fora value of =. equal either limit, the locus is infinitely small. Having remarked the principal features of these three divisions, the problem can now be considered under a different form. For as the relations of the involved data were expressed by APA 0.07: Bet Cd tam ln they are evidently expressed by nm (AP?) — m (BP?) = m (c.d) — n (a.6) and this can be orally expressed te the two following manners; just according as we suppose — ~ positive or negative, Viz :— First enunciation—(when = is positive) —“ Given two points A and B in position ; to find the locus of a point P, such that n times the square of its distance from A, minus m times the square of its distance from B shall be equal to a given magni- tude g.h,” regarding n and m as real numbers. Second enunciation—(when — is negative) — Given two points A and B; to find the locus of a point P, such that n times the square of its distance from A , plus m times the square of its distance from B, shall be equal to a given magnitude g.h,” regarding » and m as real numbers. It is evident we can always determine a. and c.d so that m.(ced) —n (ab) = gh, or, better still, we can always find a.d ane that —n (a.6) = gh, and then ‘regarding cd = zero, we still have m(c.d) —n(a.b) = g.h, and ZN ence 77 2 BP? :: m:n. In this form it is evident (from the preceding part of the discussion) that one of the poimts f coincides with B, and that the other point f satisfies the relation fA.AB = ab. Now, when = is negative (and O between A and B) as we can always assume m negative, and 7 positive, it is evident that as n(AP?—) m(BP*) consists of two necessarily positive terms, the y.4 in the second of the two preceding enuncia- tions, can be regarded as always necessarily positive. This and Loci of Apollonius, &c. Fai | being done, it follows that the a.6 for the second enunciation is negative, and .*. according as AO or . AB is less than, a.b equal to, or greater than Af or or em so will the locus be real, real and infinitely small, or imaginary ; or which amounts to the same thing—according as ™. AB? is less than, equal to, m-%% or greater than g.h, so will the locus be real, real and infinitely small, or imaginary. Again, when “* is positive (and therefore O is in a production of AB) according as g./ in the first of these two enunciations is negative or positive, we can take m and n, both positive or both negative. Doing this makes the corresponding a.6 always positive, and gives the point f which does not coincide with B always in the production of AB through A. Hence we see that the locus is always real in this case when 7 is nume- rically greater than x. But when m is numerically less than n, it is evident that according as AO or ()AB is longer m= b -@l than fA (or = or —%,), or equal to it, or less than it, so will the locus be real—real and infinitely small—or imaginary; or, which amounts to the same thing, according (“ )AB? is in extent greater than—equal to—or less than g., so will the locus be real—real and infinitely small —or imaginary. NOTES. In respect to the general loci problem just investigated, it may be remarked that if S be the point in which CD cuts AB, then will AS : BS :: AC.PD : BD.PC. And since this last ratio is evidently compounded of the known ratios, AC.AP to BD.BP, and PD.PB to PC.PA, it follows that the point S is known, as also the magnitude of the rectangle SC.SD. And from this it is evident the problem may have had its origin in the following GENUINE* ANCIENT PORISM. If the sides ab, be, ed, de, ef, fg, of a polygon abcdefg in- * The ancient porisms were formal investigations for theorems, and for the relative states of the data of problems which might cause the solu- tions to be innumerable. The modern porisms are but theorems originally derived from porismatic researches. 78 The Three Sections, Tangencies, scriptable in a circle pass through fixed pois A, B, C, D, K, F, G, and that the rectangles Aa.Ab, Bb.Be, &c., under the distances of the fixed points from the angular points on the sides through them are constants ; to find the relations amongst the data so that the locus of a, any angle of the polygon, may be found, &c. ANALYSIS. Let O be the centre of the circle abcdefg. Then, since AO? —rad2.O and GO? — rad? O are equals to Aa.Ad and . Gg.Ga, ... GO? — AO? must be constant, and the perpendicu- lar from O on AG must cut it ina fixed poimt. Similarly, the perpendicular from O on BG must cut it also in a fixed poimt. And hence we see that when any three A, G, B, of the points are not ina straight line, the centre O of the circle is fixed, as also the circle O itself. And in this case the locus of a must be—if anywhere—in the circumference of the circle O. But if the fixed points A, B, C, &c., be im a straight line, it is evident the centre O is not a fixed point, and that its locus is restricted to a straight line perpendicular to the line ABC, &c., and that its circumference must pass through two fixed points, real or imaginary, in the lime ABC, &e. More- over; it is evident that if we put X for the pomt m which the perpendicular from O cuts AB, &c., and that we put 2”, a’, 6’, c’, &e., for the distances of X, A, B, C, &c., from an assumed point on ABC, &c., then must the following relations exist, ViZ.0'(a —2')? (62) = lm, (a =e) = (C=) ae (d—2')) — (d'-a')) =I-p, (a'-2')) — (¢-a') =1-g, (a= gw)? — (fiH2eeal=r, (=a? = 9-2) = sine l, m,n, p, 7; 7, 8, represent the magnitudes of the constant rectangles. And determining on any N+2 of the 2N+1 quantities represented we can easily find the remaining N—]. Now looking on the three consecutive sides ab, bc, cd, of the polygon, we know (from involution property of mscriptable quadrilateral) that when the points A, B, C, D, EH, F, G are in a straight line, ad passes through a fixed point im this hne; and lookmg on the three consecutive sides ad, de, and ef, we know that af passes through a fixed point in the same line ABC, &c. And in this manner we may evidently proceed until we come to the last so formed quadrilateral having its fourth side coincident with aside of the polygon, or infinitely small, and tangent at a, just according as the number of sides and Loci of Apollonius, &c. 79 of the polygon is even or odd. Now the points and magni- tudes being as above indicated, it is evident that when the number of sides is odd, the locus of @ is a circle having as centre the fixed pomt in which the tangent at aw cuts ABC, and a radius equal this tangent, whose magnitude is evidently constant. And when the number of sides of the polygon is even, the locus, under the prescribed necessary conditions for inscriptable poly gons, is not restricted in the plane. Again, returning to the case in which the points are not in a straight line, we have seen that the circle O is fixed, and that the locus of a—if ais at all capable of innumerable con- tinuous positions—must be the circumference of circle O. And here, in order to facilitate the investigation of the relations, let us first suppose a polygon of three sides, and see whether it is possible to have three poimts A, B, C not ina straight line, so related that if through any one of them C we draw any chord rs to a fixed circle, and Ar to cut the circle again in /, and As to cut the circle again in v, then will sf and 7v pass through B. Now, from the properties of poles and polars, it is easy to see that when each of the three points is the pole of the line joiming the other two, these conditions will be fulfilled; more- over, since if any two A, B of the pomts so determined be fixed, the third C is in the intersection of the perpendiculars from A on OB and from B on OA, it follows that the given poimts must be restricted to the number three when they are not all in a straight line. I need scarcely mention that similar investigations may be made when instead of points in a plane, &c., we are proposed points on the surface of a sphere, and the products of the tan- gents of the halves of the segments made on the sides of the inseriptable polygon by the fixed poimts through which they pass; or, when we are proposed points in space, and a figure inseriptable im a sphere, and constant rectangles under the distances from the fixed points to its angles situated on the sides passing through these fixed points. PORISMATIC RESEARCHES. (The figures to be supplied by the reader.) Given three straight lines MM, NN, and LL, in position, and given the point P in the first line MM, and the point Qin 80 The Three Sections, Tangencies, at the second line NN ; through two given points B, C, to draw two straight lines BI, Cl, intersecting each other on Li, the third given line, so that E and F being the respective points in which BI and CI cut MM and NN, we shall have PE to QF in the given ratio of m ton. (The magnitude of = is not only considered known, but its sign also—the directions on MM and NN being considered.) ANALYSIS. Se -From porisms in the Transactions for 1859, we know that if we draw QA making the angle QN right to A =PM right to B, and that we make QA to PB as QF to PE or as m : n, then will BE and AF intersect in a point O in the circumference of the circle described through A, B, and the point R of intersection of PB and QA. eal And from porism 8 in the Transactions for 1859, we know that if we draw CG and CD parallels to NN and LL to cut LL and NN in G and D, then will GI.DF be equal to the © known magnitude GC.DC. ‘Therefore, since the pomts B and A are known, and also the circle ABR, it follows (by problem second, Jransactions for 1859) we can draw the straight lnes BOI and AOF, and therefore also CIF through I where BO cuts LL. COMPOSITION. Through C draw CG and CD parallels respectively to NN and LL, to cut Land NN in G and D; draw QA making the angle QN right to A equal to PM right to B, and make QA : PB:: m: m, describe a circle through A, B, and R the intersection of PB and QA; find a poimt O in the circle’s circumference (by problem second, Transactions for 1859) such that I and F being the points in which BO and AO cut LL and NN, we shall have GI.DF = GC.DC; draw FC to cut LL, then will BI and FC be answerable lines. . For let E be the point inwhich BI cuts MM. _ By porisms, _ (in Transactions for 1859) we have the poimt I on the line PC; and PH: QB: sania. PORISMATIC RELATIONS OF DATA. % Before entering on the investigation of the porismatic _ relations, it may be as well to observe that it is evident that and Loci of Apollonius, &c. 81 as there are two answerable pairs of lines BO, AO (see problem second, in Transactions for 1859), there are two pairs of answerable lines BI, CI; and that according as the lines BO, AO are real and imaginary, so will BI CI be xreabor imaginary. Moreover, it is evident that when + is not restricted as to sign, there are four solutions to this problem, two of which cannot be always real, &c. Now, it is evident (from porismatic theorems in Transactions for 1859), that in the porismatic state of the data of the present problem, we must have the angle DN rightto A = angle GB right to L,and DA.GB = DC.GC; and, moreover, a straight line through A parallel to NN must cut BG in a point S on the circumference of circle ABR. But (from porisms in Transactions for 1859) the angle OA right to B is equal angle NN right to MM, and .-. angle SA right to B being equal to this last angle, it follows that SGB must be parallel to MM. Again, for any finite value of “, it is evident that QC must pass through the pomt H in which BP cuts LL. It is also evident the triangles BGC, CDA, are similar. And if T be the point in which DA cuts the circle ABOSR, we have the angle BS right to T = AS right to T = DN right to A = GB right to L; and therefore BT is parallel to LL. Hence, we see that in the porismatic state QC and PB intersect on the line LL, and that BG and CG parallels respectively to MM and NN intersect on LL, and that the ratio ~ is equal to the ratio of the segments PU and QD, made on MM and NN by straight lines BU and CD parallel to LL. ald , = And we deduce the following important porisms :— PORISM. Given two points B, C, and two straight lines MM, NN, in position, and the point P in MM, and the point Qin NN; a _ straight line LL can be found, such that I being any point in™ it, and & and ¥ the respective points in which 1B and IC cut MM and NN, we shall have PE to QF in a constant deter- - minable ratio. . For BG and CG parallels to MM and NN, give us one point G inthe required line ; and QC and BP give us another point in the required line by their intersection: therefore the G 82 The Three Sections, Tangencies, line LL through G and H isknown. The determinable ratio can be found by drawing CD and BU parvallel to LL, to cut NN and MM in D and U: itis that of PU to QD. PORISM. Given the point P in a given straight ine MM, and a point Qin a given straight line NN, given also a point B, and a straight line LL; a point C can be determined such that lines drawn from any point I in the third given line Li, through the given point B and the determined point C to cut MM and NN, respectively in EH and F, we shall have PE to QF ma constant determinable ratio. For BPH is known, and BG parallel to MM is known, and therefore GC through G parallel to NN is known in position; and hence the point C where GC cuts QH. And I being any assumed point in LL, and H, F, the points in which BI and CI cut MM and NN, the required ratio is evidently the same of that of PH to QF. Or, drawing CD and BU parallel to LL to cut NN and MM in D and can the required ratio is evidently the same as that of PU to QD (the lines CD, BU intersect at infinity on LL). PORISM. Two points B, C, and two straight lines MM, LL, are given in position, and a point P inthe first of these lines MM ; a straight line NN and a point Q in tt can be found, such that if lines 1B and IC be drawn from any point I in the second LL of the given lines through B and C, to cut MM and NN, in EK and F respectively, then will PE have to QF the given ratio of 1 to un. For we can draw BG parallel to MM to cut LL in G, and therefore we have CG to which NN must be parallel. Again we have BP and point H where it cuts LL; and therefore HCQ through H and C is known in position. And we have *PU, U being the point in which BU parallel to LL cuts MM. And if D be the point in which CD parallel to LL cuts NN,- then sinceQD: PU :: m:n, and that QDis parallel to CG, it follows QD is known, and therefore NN with which it is coincident. If the ratio = be not restricted in sign in respect to direc- tions on CG anid MM, it is evident there are two positions and Loci of Apollonius, &e. 83 for QD or NN equidistant from C and parallel to CG; and ‘. also two points Q, one in each line NN. PORISM. Given the lines MM, NN, and the points P and Q therein —one in each line; given also a point C and a ratio = 2: innu- merable points B and corresponding straight lines LL parallel to any given direction can be Sound, such that I being any point in LL and i, F, the points in wheel = and IC cut MM and NN, we shall have PE : QF : n. For QCH is known in PE eS we CG parallel to NN is known in position, and CD parallel to the given direction is known in position, and DA making the angle DA right to C = NN right to MM is known in position ; and the circle QPR is known. And since PU :: QD:: m : n, the point U is known, and hence UB parallel to CD is known. And R being any point in the eircle which passes through the inter- _ section of MM and NN, and through the points P and Q, the lines RP and RQ will cut UB and DA in B and A, making the triangles DAQ, UBP similar ; therefore drawing BG parallel to MM to cut CG in G, we have the triangles BGH similar to QDA, and therefore LL through G parallel to CD is a required line Lh, and B is its corresponding point (the locus of B is evidently a known straight line UB). If = be not restricted as to sign, then evidently the entire locus of B is two known parallels equidistant from P, &c., &e. PORISM. Given two points B, C, and two lines MM, NN, in position, and a point P in MM; a point Q can be found in NN, such that straight lines BI, Cl, to any point I in a determinable straight line LL shall cut MM and NN in E and F, so that PE shall have to QF the given ratio of m to n. For we have BG and CG parallel to MM and NN; and therefore as the triangle CDA is similar to BGC, the locus of A is a known straight line KK. And as QA lies between NN, KK, making the angle QA right to N = PB right to M, and that it has to PB the given ration : m, .. QA is evidently known in magnitude and position ; hence, the line CD (making the angle CD right to A = BG right to vi is known, and .*. LL through G ee allel to CD. G2 ae 84.7% >» The Three Sections, Tangencies, If the ratio — be not restricted in sign, it 1s evident there are two positions for QA equally distant from the intersection of KK and NN, and therefore also two answerable lines LL corresponding to the points Q and Q. oe # PROBLEM II. Given three straight lines MM, NN, and LL in position; and the points P and Q in the two first lines—one in each; through two given points B, C, io draw straight lines BI, Cl, intersecting each other on the third g given line, so that B and ¥ being the points in which they cut MM and NN (the first and second lines) we shall have PH.QFE equal a given magnitude m.n(the sign of m.n is supposed known, &c.). ANALYSIS. From porism 10 in the Transactions for 1859, it is obvious that if we draw QA making the angle QN nght to A = the angle PB right to M, and QA.PB = QF.PE = m.n, and that _ through A we draw a parallel to NN to cut PB im K, then % will FA and EB intersect im a point O in the circumference of the circle ABK. And from porism 8, in the Transactions for 1859, it is evi- dent that by drawing CG and CD parallels respectively to NN and LL, to cut ‘Tal and NN in G and D, we will have GLDF = GC.DC = a known magnitude. And since © through A and B, the les AO and BO ‘(ntersecting in the known circumference AKB), make GI.DF of a known mag- nitude; therefore (by the,second problem in the Transactions for 1859) AO and BO are known in position, ng hence the points E and F, and line FCI, &c. oe P REMARKS. The Composition and Discussion may be easily made if thought» necessary, but as my chief object is to develop the porismatic relations of the involved data, I will at once pro- ceed to do so. ' W. Porismatic Relations of Data. It is evident, from porisms in the Transactions for 1859, that this problem will become. poxismatic when the angle GL right to B = DA right to N, and that GB.DA = GC.D And if K’ be the point in which GB euts the circle AB ” «“ te? Be : & and Loci of Apollonius, &c. 85 ¥ K’A must be parallel to NN, because the segment GI corre- sponding is infinitely small. Therefore in the porismatic state of the data, PBGK must be a straight line. And the point Q must be such that after drawing DA making the angle DA right to N = GL right to B, and DA.GB = DC. GC, we shall have the angle QN right to A = PB right to M, ‘and QA.PB = mn. ‘And for similar reasons to those which cause PB to pass through G, will the straight line QC pass through ie point H in which BH parallel to MM euts LL. Hence ) we easily deduce the following important porism . “f ™" PORISM. A point P (and but one) can be found in a given straight line MM, and a. corresponding point Q in another given line NN, such that if straight lines BI, CI, be drawn from any point 1 in a third given line LL thr ‘ough two gwen points B, C, to cut MM and NN respectively in E and F, we shall have PE.QF of a constant determinable magnitude. For (from what has been observed) drawing BH and CG parallels respectively to MM and NN, to cut LL in H and G, it follows that GB will cut MM in P, and HC will cut NNinQ@. And drawing CD and BU parallels to LL, to cut NW and MM vespectively in D and U, it is evident (since _ these parallels intersect at infinity on LL) PE.QF must always = a 4 ; PORISM. Given in position a straight line Lb and the points Band C; through two other given points P, Q, two straight lines MM and NN«(and but two) can be dr awn, such that 1 being any point in LL and E and F the points in which 1B and IC cut MM and NN, we shall have PE.QF of a constant determin- able magnitude. * ~ _ For the lines PB, and QC give us the sicamitn Gand Hin their intersections with LL; and PMM, QNN, must be parallels (respectively) ‘to BH and GC. And PE.QF = PU.QD = a known magnitude, PORISM, a Two points B, C, and two'straight lines MM, LU, are given ei in position, and a point Pa in MM; a straight Tine NN and a : & ed 86 The Three Sections, Tangencies, point Q in it can be found, such that 1 being any point in LL, and BK, F the points in which IB, IC cut MM and NN, then will PE.QF be of a given magnitude m.n. For PBG, BU, and BH are known, as also HC; and CD parallel LL is known in position; and .-. since PU.QD = m.n, it is evident NDQN between HC and CD and parallel to CG is known in position. If m.n is not restricted in sign, there are two answerable lines NN equidistant from CG, &c. . PORISM. Given the points P, Q, in the gwen lines MM, NN, and given also a point B and a straight line LL in position ; another point C can be found, such that I being any point in LL, and E and F the points in which 1B and IC cut MM and NN, we shall have PH.QE of a constant determinable magni- tude. For PBG, BH, and HQ are known, and therefore the point C where. GC parallel to NN cuts HQ, and therefore also PE.QF = PU.QD is determined. PORISM. Given the points P and Q in the given straight lines MM and NN, and given also a point B; a point C and a straight line LL, parallel to a given direction can be found, such that I being any point whatever in this line, and KH, F the points in which IB and IC cut MM and NN, we shall have PH.QF of a given magnitude m.n. The straight line PBG is known in position, and the line BH parallel to MM is known in position; and QA and point A are known ; and drawing BU parallel to the given direc- tion, the point U in MM is known, and therefore D in NN (because PU.QD = PB.QA = m.n) ; therefore DC parallel to BU is known in position. And since if through A we draw AD’ parallel to CD (since angle AC right to D’ = CA right to D = BCright to G) the circle ACB is known ; hence the pot C in which this circle cuts DC is known, and therefore CG parallel to NN, and the point G in which it cuts PB, and therefore LL through G parallel to DC or BU. Itis evident there are two answerable points C, and that to each one corresponds an answerable line LL. | cf and Loci of Apollonius, &c. 87 If the sign of m.z is unrestricted, there are four answerable points and their corresponding lines LL, &c., &e. NOTES. By means of the porisms now given, and those in the Transactions for 1859, we are enabled to give simple solutions to the following interesting problems, viz. :— To construct a quadrilateral having its four sides passing through given points, and two of its opposite angles on given straight lines, and such moreover that the other two angles shall be of given magnitudes, or be situated on given straight lines, or such that one of them shall be of a given magnitude and the other be situated on a given straight line. They can be easily reduced to the third problem i in Trans- actions for 1859. And from the porismatic state of the data in these problems numerous interesting and useful porisms may be derived— ~ one of which is as follows :— Given a point P and two straight lines MM, NN; then if B and C be any points in these lines such that the angle PB right to C shall be of a given magnitude, another point Q may be found such that the angle QB right to C shall be of @ con- stant determinable magnitude. - FINAL OBSERVATIONS. In the preliminary observations I have pointed out advan- tages to be derived from my new methods of defining angular magnitudes ; and those who have carefully studied the solu- tions to the problems contained in this paper, cannot bé unaware of the value of these advantages. However, I will now give a few other instances of the improvements which they effect in important theorems :— First. The theorem on page 465 of the Géométrie Supé- rieure should be replaced by the following :— A moveable tangent tt rolling on the cir cumference of ae circle whose centre is C, cuts two fixed tangents m'm’. n'n’, in two points m, un, such that the “angle Cm right to n” as constant = 9 right. Second. The theorem on page 457 (which is so defective that its reverse is untrue) should be replaced by the follow- ing :— 4 3 ak <. a - 88 The Three Sections, Tangencies, If A, B be two fixed points in the circumference of a circle, and P any point whatever in the circumference, then will the angle PA right to B be constant = 6 right. And I may remark that by adopting this property as the fundamental definition of a circle, it would be easy to frame ‘an Element of Geometry in which we could give a direct demonstration to the fact, that a point O, and but one, exists from which lines drawn to the circumference of the circle (or locus of P) are equal to each other. é . © This might be done in various ways ; but perhaps the most simple, as well as the most natural,* would be to have a -knowledge of the Theory of Numbers, including proportion in its most general form, precede the study of geometrical science, so as to establish the relations of straight-lined figures as early as possible. The demonstration might be made as follows :— Bisect AB in C and through C draw MM perpendicular to AB; find O in MM such that the angle OC right to B= 0 right ; then will O be the sought point. For let I be the other point in which the circle described from C as centre with CA or CB as radius again cuts AP. re ' mi * On careful examination it will be found that all our Geometry rests ultimately on our power of conceiving the positions and motions of points, lines, and surfaces, and on acquirements in the Theory of Numbers, whether we depend on our intuitive ideas of these things, or that we acquire a more extended systematic knowledge before commencing the study of a regular course of geometry. And this should lead us to look on the Theory of Numbers as primary to geometry, inasmuch as without some knowledge of numbers there can be no systematic treatise, and that the introduction of it from the commencement confers simplicity and comprehensiveness on the Elements. However, many are to be found who think they proceed with- out the use of numbers or proportion, when they disguise their investigations so as to mislead themselves: for wherever there is an idea of equality there must be an idea of proportion; though not necessarily so refined or compre- hensive a one as Euclid’s. Others are to be found who would trammel the natural action of the intellect, and compel the geometer to discard all con- ceptions of motion, as foreign to the spirit of pure geometrical science. They point confidingly to what works of the ‘‘Ancients” now remain in support of their crotchet, asaf the philosophy of the Greeks were perfect, and should prescribe limits to the march of progress. But, though truth may be for a * time suppressed, by the combined efforts of masters and professors it will yet triumph ; and an axiomatic motional philosophy will correct the faulty links and defective logic in the system of geometry which now unfortunately preyails. Indeed, as in other sciences, it will be found, on careful examina- tion, that geometry has inherent properties independent of man’s endow- ments, and exacts peculiar mental operations, from capable beings who fathom some of its various truths, which cannot be prescribed or varied at man’s” pleasure. for they are ordinances of the laws of nature, and can change only, at the will of Providence. ~ and Loci of Apollonius, &e. ‘ 89 The triangles BCI, ICA being isosceles, we have the angle IA or IP right to B = CB right to O; but PA or PI right B = OC right to B; therefore the angle BC right to O = angle BI right to P, and the equiangular triangles OCB, PIB give BC : BI: : BO: BP. ; And since the angle BC right toO = BI right to P, we have the angle BC right to. I = BO right to P; and, there- fore, since BC : BI :: BO: BP, the triangles BCI, BOP, are similar, and give OB : OP :: CB: CI. Hence, as CB = CI we have OB = OP bb. COROLLARY. If P’ be the position of P infinitely near to or coincident with B, then P’B is tangent at B; and putting MM for this tangent, we have angle BA right to M = PA right to B = OC right to B, and .*. BM is perpendicular to BO. - .. = oe he ss ay Art. I1.—A Communication from the Reverend W. B. CxiarkeE, of Sydney, to His Excellency Sir Henry Bark y, K.C.B., &c., &e., President of the Royal Society of Vic- toria, on Professor McCoy’s ‘‘ New Teniopteris” from the Coal-bearing Rocks of the Cape Paterson District in parti- cular, and on the Evidence bearing on the Question of the Age of Australian Coal Beds in general.—Communicated to the Society by His Excellency THe PresipEnt. [Read before the Royal Society, 25th June, 1860.] St. Leonard’s, 1st June, 1860. My Dear Sir, . I recetvep with much thankfulness the kind reply which you were good enough to send to the letter which I did myself the honour of addressing to your Excellency. The information contained in it respecting the discovery of Teniopteris, at Cape Paterson, is very interesting. But I hel 90 Rev. W. B. Clarke’s Communication on hesitate, at present, to admit that such a discovery determines the fact of the existence of the Jurassic formation in Victoria. Of course, not knowing what species have been found, or even if the genus be really Teniopteris (for many of those plants so called are not Teniopteris at all), it is impossible for me to come to any conclusion on the subject. But in a paper which I propose to write, so soon as I can find leisure, I will en- deavour to show what are the real grounds upon which I have ventured to contend, and still do contend, against the sweep- ing assertion of those geologists who maintain that a formation so abundant in zoological fossils (more so than, perhaps, any other) as the Jurassic, is found here, where no one, in any part of the Australian continent, has ever detected one single species, on the strength of the evidence derived from a few. (probably not six in all) species of plants, the true description of which does not agree in all things with the typical charac- ters of the genera under which the species are ranked. Thetwo genera, Teniopteris and Glossopteris (Sagenopteris), have been the means of placing, by some geologists, the coal deposits of Australia and India in the horizon of the oolitic coal. Now the latter occurs in no less than five distinct formations in India, as Mr. Oldham informs me, and it also occurs in Africa, where the evidence appears to be against the supposed epoch. As to Teniopteris, so far from the genus determining the age of a formation, Jukes, who follows Bronn, assigns the species thus :— Carboniferous 1] Permian .. Me ses ie Trias. os. ys a i ee Oolitic ae 2 iO Tertiary .. ae ei It is, therefore, the species which must determine whether the new found plants belong to the oolites or not; and when we come to Yorkshire, which is one of the references, we find in Phillips no figure of any species of Teniopteris, and only one catalogued 7’. latifolia, of which he gives as synonym 7. major of Lindley and Hutton, which is not a Teniopteris at all, ze. if we are to regard Brongniart’s description of the genus as that to which we are, undoubtedly, to have respect. Again, Morris assigns to 7. major the synonym » 9 | Euston, 22 miles below last place .............06 30°12 5:00 3», 4 | Dry Lake, 8 miles E.N.E. from Euston ...... 30°15 13-00 » 9 | Mr. Ross's Station, Mailman ................00. 30°15 11:00 Bs 5 eens -ridge, in Mallee Scrub, ae sOUNDsMsncewecees 30°16 24-00 6 dge of open country, 30 miles N E. from . ; ty “Mailman.........:.... PIG CRATE 30°20 4°67 Bs 6 | Tacnall, 38 miles N.N.E. from Mailman ...... 30°21 18:33 43 7 | Head Winter alas (Nit RGSSts) wets csenceo tes 30°14 2°50 11 | Camp, 30 miles of last place, or of {Qc : a latitude 34°17! 44 oss csscccssssee it ae As », 12 | 10 miles E., 30°S., towards Murrumbidgee, \ 29-94 28-00 Mets aver Mawmtidens ce ok a eat 29:87 2°75 5, 15 | Above junction of. River Lachlan .......,....... 29°82 Zero 53 eolGiy|pOxley Reserve. J. .cjccucsecsssteneoh codensseacenreaces 29°47 9°75 >) 18 | 21 miles above; Oxley, i.)i2..:sessssencdeneectaceseence 29°52 15°50 >, 19 | Waijeers, 41 miles above Oxley ...........cs0c00 29°56 7:50 >» 20 | Hurst’s Station, 20 miles above Waljeers...... 29°76 5°50 53 a Anabranch of Lachlan, 16 miles above Huet s 29°69 2°25 Z 18 miles above last place and 14 above Sul- ; s 4 lar and Bowler’s sernsene sens So ocee tees \ 29°46 14:50 23 |7 miles above last place, and 23 above , : “s aay (raining) Percent 29°47 15°75 26 | 11 miles N.W. from Lachlan, on sand hill 3 : a BOfeet high .....cs.cccsse-e. are teary ties t 29°65 9:00 >, 27 | Large creek, 30 miles N.W. from Lachlan... 29-74 3°50 », 28 | Windmill Swamp, 10 miles N.W. aa 29-84 0:50 TAY OO ICLOC Keen cuversacscetianecedecseceenaets below Zero AS Peo Casuarina achiurnyp-red..ceccndusesmsceteceaemene a ke ses 29-94 2°00 eee cas point reached, about 5 miles from 30-40 18-00 ASb eee vacation ace ete atenueweetamaesumltroeteec tees dl | Camp 13 miles 8. from where the large ; 3 ce creek was first crossed............sseesseeeses 29°84 1°00 June 1 | Camp by a dry creek, about 8 miles N. W. from Anabranch of Lachlan, and sup- 29-70 300 posed to be the same as that crossed 7 below Zero miles N.W. from Ramsay’s » 8 | Suttor’s Station, on Anabranch gett 29°57 2:00 soy iy) LOxley:s: Marked, Teens. co® swancevaghacen senaeeeie 29°50 11°75 >» @¢ | Sand hill, 5 miles B. of Waljeers ..........0.+ 2948 6:50 > 8. | Hay, on Murrumbidgee.......:.....deccsssssveeseoss 29°40 4°50 >» 9 | Old Man Plain, 8. of Murrumbidgee ............ 29°50 208 a : 1:50 Ary kd) Billibong Creoketera.cn, coh antec iseetenenacmnsenscte cence 29°64 nelow Vex >», ll | 12 miles N. of Deniliquin................ at sess 29-54 7:00. 11 | Deniliquin (sunset)..,......... Sacainaeaan ea cie Doe 29°51 14:00 On the Nest and Eggs of the Coach-whip Bird, §c. 141 Art. VI.—On the Nest and Eggs of the Coach-whip Bird (Psophodes Crepitans, Gould), and of the White-fronted Epthianura (Epthianura Albifrons, Gould), with some gene- ral remarks on the Nidification of Australian Birds. By A. Dosrez, Esa. [Read before the Royal Society, August 27, 1861.] As neither Mr. Gould’s work on Australian ornithology, nor such others as I have been able to consult, contain any par- ticulars of the nidification of the two above-mentioned species, I am led to the conclusion that, although probably casually discovered, no description has as yet been published, and beg therefore to communicate the following notes, from personal observation. lst. Psophodes Crepitans (Gould). Coach-whip Bird. This bird is well known to most Australian colonists, though, probably, far more frequently by the peculiar note whence it derives its current name, than by its appearance, as it generally keeps itself concealed amidstthicketsand brushes. To naturalists this species is extremely interesting, as its sin- gular characteristics have long rendered it a matter of doubt what precise place to assign to it in the ranks of Australian birds. The details of its nidification may tend to the further elucidation of this point. It will not be out of place first to quote a few of Mr. Gould’s very accurate remarks on the habits of this bird. He says, “Tt is only to be found in dense brushes, and is a shy and recluse species; for although its full notes—ending sharply like the crack of a whip—indicate its presence, it rarely ex- poses itself to view, but generally keeps in the midst of the densest foliage and among the thickest climbing plants, fre- quenting alike those that have intertwined themselves with the branches of the tallest shrubs, and those that form almost impenetrable masses near the ground, and through which it threads its way with the utmost ease. It is extremely ani- mated and sprightly in all its actions. Of its nidification, I regret to say I know nothing, although I paid great attention to the subject myself, and offered rewards for its nest and eggs, and for any information respecting them.” 142 On the Nest and Eggs of the Coach-whip Bird, &c. The present nest and eggs were obtained by me near the banks of the Yarra Yarra, near Heidelberg, on one of those points of land or ‘‘bends” of the river still left in their original state, and where the underwood and tangle are extremely dense. Being on a visit in the neighbourhood on the opposite side of the river late in the previous summer, my attention was attracted by the remarkable note of this bird, but as the breeding season was then past, I merely noted its haunt. Finding, however, on reference to Gould’s work, that its nidi- fication was unknown, and relying on a general fact I had - often observed, namely, that the same pair (apparently) of birds will, if undisturbed, return for several years to the same locality for breeding, I revisited the spot about the com- mencement of the next summer, and, after a short search, was rewarded by the discovery of the nest, on which the female bird was sitting so closely as almost to allow herself to be captured; thus removing all doubt as to the identity of the nest and eggs. The nest was in the most tangled part of the thicket, and placed in the forked branches of a shrub, about four feet from the ground—it is cupshaped, about five inches outside diameter, the exterior of dry slender twigs, and the interior lined with thin fibres and a few pieces of horse- hair, the latter evidently owing to the accidental vicinity of some farms; the whole structure is neither very solidly nor elaborately built. It contained two eggs—tlength, exactly one inch; extreme width, three quarters of aninch. In shape they are not much pointed at the thinner end, and the greatest girth is at about the middle. Their ground-colour is pale greenish blue, with streaks and dots of various sizes scattered pretty equally over the whole surface ; these markings are of a brownish black colour, and of two kinds, the one being very distinct and sharp, the other somewhat less numerous, more greyish, and much fainter, having the appearance of being under the shell. From the fact of the bird sitting so closely, I conclude that no more than two eggs are generally laid, though the present ones had not yet been perceptibly in- cubated. I regret to say I have kept no precise memorandum as to the date of finding the nest, but believe it to have been about the end of October. Qnd. Epthianura Albifrons. Whaite-fronied Epthianura. This bird, which at the first cursory glance recals the familiar English black and white Water-Wagtail, both by its On the Nest and Eggs of the Coach-whip Bird, &c. 148 general appearance and motions, is by no means uncommon round Melbourne. Mr. Gould writes as follows in his notice of this species :— “JT first met with it in a state of nature on the small islands in Bass’ Straits, where it had evidently been breeding, as I observed several old nests in the barilla and other stunted shrubs—its natural province is the ground, to which it habi- tually resorts, and decidedly evinces a preference for spots of a sterile and barren character; it trips along with amazing swiftness, with a motion that can neither be described as a hop or a run, but something between the two, accompanied by a bobbing action of the tail. Of its nidification I regret to say nothing is at present known.” It may be met with in the dry portion of the swamps ex- tending between the Saltwater and Yarra rivers. I discovered its nest about four feet from the ground, in a stunted bush, on the edge of the dense “ teatree” scrub which covers part of that locality. The structure is cupshaped, somewhat deep, and about four inches outside diameter; dried fibres, fine twigs, and stalks form the exterior, and the lining is composed of horsehair and fine grasses. It contained three fresh-laid eggs; length, 11-16ths inch; extreme width, 17-32nds inch ; shape, not much pointed ; ground-colour, white, with fine red- brown markings, consisting of points, streaks, and roundish dots, the larger markings being most abundant at the thicker end, where they form a sort of wreath, while some of the smaller ones are scattered over the other parts of the surface. The markings are, in nearly every case, surrounded by a faint ashy margin of their own colour, imitating the appearance of their having been painted on the white ground before the latter had properly dried, thus causing them partially to run into the white surface. This seems to be a decided charac- teristic in these eggs. The nest was discovered about October. Mr. Dobrée then proceeded to make some general remarks on the most interesting forms of nidification of Australian birds, in which respect, he stated, this country maintained its reputation for singularity. He alluded to the mound-raising Leipoa, or mallee-scrub pheasant, an egg of which he ex- hibited ; the Yellow-tailed Acanthiza, of whose singular double-roomed pen- dant nest a specimen was shown ; and remarked on the burrowing habits of the Paradolotus, as well as the hanging structure of the Yellow-throated Sericornis, which he produced for inspection, he also recurred to the fact of the Australian representatives of the Cuckoo-family, though deprived of the familiar note, differing in no way from their European cousins in the habit of confiding their progeny to foster-parents. He further exhibited a valuable collection of Australian eggs, including that of the Lyre-bird (Menura superba) of which hitherto only a limited number of specimens have been 144 On the Nest and Eggs of the Coach-whip Bird, Sc. obtained; and pointed out that many further observations were still desirable for the completeness of information in this branch of natural history. He remarked, in conclusion— One of the secondary causes of the often noticed scarcity of birds in Australia, as compared with England, is undoubtedly that so many species here lay a much less number of eggs. Amongst the commoner of Australian birds, the Honeyeaters (Mcliphage) average 2, and some species only one solitary egg; the Watilebirds (Antochere) 2 or 3; the Sericornis tribe 8; the Fosterops 3; the Woodswallows (Artamis) 4, &c., and this opinion is corroborated by the fact that the few ex- ceptional species whose individuals are really numerous, such as the common quail (Cotarnix pectorelis) and the Parokeet tribe are found to be large layers, the quail producing 11 to 14, and the parokeets 6 to 8 eggs. In England nearly all our commoner birds average 5, or at least 4 eggs, and there are many instances of a larger number. I said that this is a se- condary cause, for Reason at once ascends higher in the scale, and asks why it should be so arranged that only a limited average is produced? Is it that this country would, in the case of most species, not offer sufficient food of a suitable kind to maintain a greater number of individuals, and therefore the all-balancing hand of Nature has struck the present ad- justment as the true one between supply and consumption ? Or, to change the line of inquiry, can we entertain the notion of a more recent origin of Australia, and thence deduct that the scarcity of animal life is in some degree owing to the fact that the same amount of time has not elapsed as in the older countries to enable this continent to be stocked up to its full capability ? However this latter may be, it is certain that the rate of reproduction, as represented in the imstance of most Australian birds, cannot be proceeding at a rate nearly equal to the annual increase in many other countries. In order to investigate fully this subject of the abundance or scarcity of animal life, it would however be necessary to take into account the different destroying causes in the countries under com- parison, as well as the reproducing ones ; and altogether the matter is one which deserves more than the few hasty glances here incidentally bestowed on it. On the Application of Galvanic Electricity, &e. 145 Art. VII—On the Application of Galvanic Electricity to Practical Astronomy. By KR. J. L. Exviery, Ese., Govern- ment Astronomer. [Read before the Royal Society, August 27, 1860.] Tue application of electro-dynamics to astronomical purposes presents one of the most marked features in the history of modern astronomy; and it is probable that without the assistance of the electric current, no improvements that could be devised in the optical construction of the instruments used would secure the almost absolute precision we can now attain in those observations in which the accurate determination of time is of primary importance. The application of electro-dynamics, styled indifferently chronography and galvanic registration, has resulted in such decided advantages in the great American and European ob- servatories, into most of which it has been introduced, that no observatory of any pretensions can be considered complete without it. The aim of my paper this evening is not so much to enter upon the history and successive improvements made in this application of electricity, as to describe to the society the manner in which it has been adopted in the Astronomical Observatory of this colony, and especially to call attention to a recently constructed chronographic apparatus, which will be of great assistance in making the astronomical observa- tions in connection with the geodetic survey. The first use of the galvanic current in astronomical ob- serving was made by Professr Locke, of the United States, as early as 1849, and although its successive adoption in various observatories has led to many improvements and modifications of the original mechanical arrangements, the principle remains the same. The general form of the appa- ratus may be described as consisting of an electro-telegraphic register, with a double system of electro magnets and indent- ing styles, each system having a separate galvanic circuit. The chronographic apparatus now in use at the Astro- nomical Observatory may be thus described :— The register, with the exception of its possessing a double L 146 On the Application of Galvanic Electricity system of electro magnets and levers, is in all respects similar to Morse’s Telegraphic Recorder, which is the instrument used in the telegraph offices m this colony. By means of a weight and regulating train of clock-work, a fillet of paper is kept moving between two small rollers at a pretty uniform rate of an inch per second; and the armatures and levers are so arranged, that, when acted on by the electro magnets, they indent the paper as it runs between the rollers in paral- lel jines, about half an mch apart. The mdenting styles are placed in a line perpendicular to the margin of the fillet, so that if the action of the two electro magnets be coincident, the two indentations will be exactly in the same pomt with relation to the length of the paper. The clock-work is wound up by a key, but there being no maintaining power, this can- not be done during the working of the apparatus. The regu- lation of the train is attained by a simple “ fanfly,” and can be stopped at any moment by means of a catch pressing against the fly arbor. The clock-work will run regularly for a quarter of an hour. The voltaic circuits, for working the electro magnets, are two: the clock circuit, and the observing circuit. The observing cireuit is thus arranged, a battery of three cells of Daniel’s or Smee’s combination being generally used for producing the currents. From one pole of the battery a wire is led to one terminal of the electro magnet of the cir- cuit, from the other terminal a wire is led to the purs of the transit instrument, and then up the side of the purs to wooden frames fitting round them, where it is connected with an in- sulated portion or anvil of a small telegraphic key; a wire attached to the cther portion, or hammer of the key, is led to the opposite pole of the battery, and thus, with the excep- tion of the interval between the hammer and the anvil of the key, the circuit is complete. A slight pressure of the finger ~ on the key will complete the ciyeuit, when the electro magnet attracting the armature, causes the style to indent the paper. The indentation, when the paper is in motion, will be in the form of a dot or a stroke, as the pressure on the key is mo- mentary or prolonged. The keys are so arranged that in whatever position he may be observing, there is always one convenient to the right hand of the observer. The clock circuit, which includes the astronomical clock, is double, in consequence of the necessity of introducing what is known as a relay magnet, whose office will be pre- sently described. to Practical Astronomy. 147 From one pole of a three-celled battery a wire is led toa terminal of the electro-magnet of the clock circuit, and a wire also connects the other terminal with the relay, where it ends ina platinum point on the top of the armature lever ; the other pole of the battery is connected directly with the relay, and ends also in a platinum point, close and opposed to the other. The circuit of this system is therefore incomplete only by the separation of the two platinum points above men- tioned, which are kept apart by means of a delicate spring. The second circuit includes the clock contact apparatus, and the delicate electro magnet of the relay, and may be thus traced :—A pole of a single cell battery is connected by a wire directly with one of two springs, which are a portion of the contact apparatus of. the astronomical clock; the other pole is connected to one terminal of the relay magnet coil, whose other terminal is connected by a wire with the second spring of the clock contact; this circuit then is only incom- plete by the separation of the two springs. The ‘clock contact” is thus arranged :—An extra wheel of sixty conical teeth is fixed in the escape wheel arbor of the astronomical clock; adelicate spring, with a jewelled pallet, is so placed that when the clock is at rest the pallet drops into the angle between the teeth; at the back of this spring is a little platinum stud; another spring, with a platinum stud, is placed just behind this, and is so adjusted that a slight lift- ing of the front spring brings the two platinum studs into contact. When the clock is going, the escape wheel revolves once in a minute, and the extra wheel moves one tooth each second, which in passing presses against the pallet, and lifts the spring just sufficiently to bring the studs into contact for an instant, and so on for each second. One of the teeth of the contact wheel is filed down, so that it passes without lifting the spring, and the seconds’ hand is so placed that it points at 60 on the dial when this short tooth comes to the pallet. The two springs are firmly but separately attached to a block of insulating material, which is itself screwed to the front plate of the clock, so that the springs are insulated from the clock works and from each other. There are screw adjustments for regulating the lift of the front spring and the distance between the studs. The object of introducing a relay magnet into the clock circuit is to avoid the oxydization of the platinum studs in the clock contact, which would result from using directly a battery L 2 148 On the Application of Galvanic Electricity sufficiently powerful to work the register magnet. The oxydi- zation would give rise to imperfect contact, and the necessity of frequently disturbing the clock works in order to re-polish the springs, which of course should be avoided if possible. — The operation of the clock circuit will now be readily un- derstood. The “contact” being made every second by the clock itself, the current from the one-celled battery causes the relay magnet to attract its armature, thus brmgmeg the two platinum points of the relay before mentioned ito con- tact; the contact completes the circuit from the battery to the clock register, whose magnet immediately attracts the armature and lever, causing the style to indent the paper. This action being repeated every second, coincidently with the oscillations of the clock pendulum, if the paper be put in motion, a series of dots about an inch apart is made on it as it passes through the rollers, every sixtieth dot being left out by reason of the short tooth in the extra wheel, this serving to mark the commencement of each minute. The following 1s the mode of observing with this instrument, as practised in our Observatory :— After seeing that the circuits are complete, the register and relay in good adjustment, and the seconds’ dots being plainlyin- dented on the paper, the observer proceeds to make his observa- tion. Let it be, for stance, the transit of a star over the wires of his transit instrument :—The instrument being in position, and the observer seated at the transit instrument, he watches for the stars’ approach to the first wire; when near it, he re- leases the detent of the clock train of the register, and the paper commences to move through the rollers indented at each second by the clock circuit. Immediately the star ap- pears behind the wire, he touches the key for an instant, and the style of the observing circuit indents the paper; he does this for each wire, until the observation is complete, when he stops the chronograph—always taking care before he stops it that a minute space has occurred on the paper during the observation, in order that the seconds may be dated; if it has not, he must let it run till one occurs; andit is usual to mark this space with the number of the minute at once, to facilitate ‘reading off,” as well as to indicate by a number or symbol the object observed. Of course, if several observations are being made in quick succession, the instrument is not stopped till the series is complete. It is the custom at the Observatory to “read off” every morning, unless the result of any observation be required at t to Practical Astronomy. 149 once, the fillet used in the previous night’s work, and with a needle point puncturing in the lines of seconds perpendi- cular and corresponding to each dot made by the observer’s circuit. The seconds’ dots are then counted from the minute spaces ; the number of the second dot preceding any puncture is noted, and the fraction represented by the position of the puncture with relation to the dot is measured by a small transparent conical scale, by which means any little irregu- larity in the length of the seconds’ spaces is rectified. The “reading off” is directly transcribed into the transit-book. The usual mode of observing without the chronographic apparatus is styled the eye-and-ear method, and a short description of it will at once make the superiority of chrono- graphy obvious. In observing by the eye-and-ear method, the clock or chronometer must be in such a position that the movements of the hands on the dial shall be visible, and the “ beats” of the pendulum or balance distinctly audibie to the observer. Shortly before the occurrence of any astronomical phenome- non, such as the transit of a star, he observes the time of the clock, “taking up”’ the beats, mentally counting them until the star passes the first wire, when he writes down on his observing slip the corresponding seconds indicated ; and if it occurs between any two consecutive seconds, he estimates as nearly as he can the fraction of a second also. For a com- plete transit, this has to be done over several wires, writing down the times between cach observation. It has been found that 15 seconds are required to observe and record for each wire, and to “take up” the “beats” again without hurry. The wires, therefore, are generally arranged so that an equatorial star shall take 15 or 16 seconds to pass from one to another. The time required to complete a transit observation over seven wires of a star having a declination of 45 degrees, is over two minutes; and the utmost precision that can be attained by the best observers, under the most favorable circumstances, in a mean of seven wires, is limited to one-tenth of a second, and it is very much to be doubted if anything near this precision is ever attained. Two different observers will seldom agree as to the exact instant of a transit or occultation, owing to what is known as personal equation, the precision of combining the visual impression of the observations with the sound of the clock- beats depending much on the age, temperament, and health of the observer. When several individuals, therefore, are 150 On the Application of Galvanic Electricity employed in an observatory, it becomes a matter of the first importance to determine their personal equations, in order that all observations may be reduced to a standard. In the chronographic method no listening to the clock is required—the clock may be in another apartment, if neces- sary—no recording by the observer durmg the observation is needed, so that the intervals between the wires may be re- duced from fifteen to three seconds ; the observer’s attention can be entirely confined to the observation itself, and is not distracted between observing, listening, and writing, and the precision attainable is at least ten times as great as by the old method. Personal equation is greatly reduced, if not eliminated. Any one acquainted with the principles of physiology will at once perceive the great superiority of the chronographic registration over the eye-and-ear method. In the latter we have two totally distinct external impressions, the view of the star crossing the field of the imstrument, and the sound of the clock-beats which require to be mentally counted at the same time; while i the method now adopted we have only the one external impression—the view of the star. The mental operation involved, in coinciding thus with the touch of the finger on the key, being so rapid as to be almost map- preciable. Again, the record on the register-paper is permanent, and in case of an error in transcribing, can be afterwards referred to; whereas there is nothing but the memory to fall back upon in case of an erroneous record by the eye-and-ear method. The amount of observing in a given time can be nearly ten times as much, and with very little labor to the observer. In cataloguing stars this becomes a great object. One clock is sufficient for any number of instruments ; and in our Observatory we use the chronograph not only for the transit but for prime, vertical, and other extra meridianal observations, by leading wires from the observing circuit to the different instruments used. Wires are also carried from the Lighthouse, where, by means of a key, observations which are being made for the astronomical ayzmuths of the different trigonometrical stations can be directly recorded on the chronographic paper, so that the otherwise troublesome and often difficult determination becomes as easy and precise as observations made in a fixed observatory. Considering the benefits derived from the adoption of chronographic registration in our permanent observatory, I to Practical Astronomy. 151 was anxious to devise some means by which it could also be adapted to the shifting observatories used in the geodetic survey. Here, large clocks bemg of course out of the ques- tion, the great difficulty was the mechanical arrangement by which the “clock contact”? could be made with an ordinary marine chronometer—a difficulty so great that, so far as I am aware, it has not been previously overcome. After several experiments I was enabled to plan an appliance, which has been admirably constructed by a watchmaker at Williams- town, and is found to work perfectly, without interfermg with the rate of the chronometer. It consists of an extra wheel, of 60 teeth, which is in the seconds’ arbor, the teeth conical, with a slight rake, and, as in the astronomical clock, one tooth is cut out as a minute data. The great delicacy required in adapting contact springs, without producing un- due friction, offered the chief obstacle ; but by placing the opposing surfaces of the teeth and pallet at the most favorable angle, and arranging so that the shghtest lift of the pallet broke contact, stead of making it, the principal difficulty was overcome. ‘The contact here is the reverse of that in the clock ; for when the pallet is in the angle between two teeth, the spring dauks on a platina stud, and the contact is com- plete ; but each passing tooth lifts it sufficiently to break contact for an instant. This peculiarity would, of course, cause a registering style to make a long stroke and a short space; but the same effect as in the “ clock contact”? can be attained by a modified relay, or reversed lever in the re- gister, The success of this appliance to a chronometer at once ad- mits of a perfectly portable chronograph being arranged, and I have had a register constructed to complete the apparatus. The fillet-paper used in the Observatory, although offering several advantages, would be very inconvenient for a portable chronograph ; the plan of the register, therefore, is different altogether to the one in the Observatory. A metal drum, six inches in diameter, and eight in length, covered with fine cloth, is made to revolve on a horizontal axis in about 35 seconds, by means of a weight and clock- work, regulated by a conical pendulum.